ตรีโกณม5-2
August 9, 2017 | Author: Hutsatorn Yenmanoch | Category: N/A
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2
øíß°å™—π µ√’‚°≥¡‘µ‘
“√–·≈–¡“µ√∞“π°“√‡√’¬π√Ÿâ ”À√—∫Àπ૬°“√‡√’¬π√Ÿâπ’È “√–∑’Ë 2 : °“√«—¥ ¡“µ√∞“π § 2.3 : „™â§«“¡√Ÿâ‡√◊ËÕßÕ—µ√“ à«πµ√’ ‚°≥¡‘µ‘·°âªí≠À“‡°’ˬ«°—∫°“√«—¥‰¥â “√–∑’Ë 4 : æ’™§≥‘µ ¡“µ√∞“π § 4.1 : Õ∏‘ ∫ “¬·≈–«‘ ‡ §√“–Àå · ∫∫√Ÿ ª (pattern) §«“¡ — ¡ æ— π ∏å ·≈– øíß°å™—πµà“ßÊ ‰¥â
º≈°“√‡√’¬π√Ÿâ∑’˧“¥À«—ß ¡’§«“¡§‘¥√«∫¬Õ¥‡°’¬Ë «°—∫øíß°å™π— µ√’ ‚°≥¡‘µ·‘ ≈–‡¢’¬π°√“ø¢Õßøíß°å™π— ∑’°Ë ”Àπ¥„Àâ ‰¥â 2. 𔧫“¡√Ÿâ‡√◊ËÕßøíß°å™—πµ√’ ‚°≥¡‘µ‘·≈–°“√ª√–¬ÿ°µå ‰ª„™â°“√·°âªí≠À“‰¥â 1.
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
88
2.1 øíß°å™—π‰´πå·≈–øíß°å™—π‚§‰´πå °“√°”Àπ¥§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘ ∑”‰¥â ‚¥¬„™â«ß°≈¡Àπ÷ËßÀπ૬ ¡’®ÿ¥»Ÿπ¬å°≈“ß∑’Ë®ÿ¥ °”‡π‘¥ √—»¡’ 1 Àπ૬ «ß°≈¡‡ªìπ°√“ø¢Õߧ«“¡ —¡æ—π∏å {(x, y) RR|x2y2 = 1} °”Àπ¥®”π«π®√‘ß (∑’µ“) «—¥√–¬–®“°®ÿ¥ (1, 0) ‰ªµ“¡ à«π‚§âߢÕß«ß°≈¡Àπ÷ËßÀπ૬ ¬“« || Àπ૬ ∂÷ß®ÿ¥ (x, y) ´÷ËßÕ¬Ÿà∫π«ß°≈¡Àπ÷ËßÀπ૬ ‚¥¬∑’Ë 1. 0 «—¥ à«π‚§âß®“°®ÿ¥ (1, 0) ‰ª„π∑‘»∑“ß∑«π‡¢Á¡π“Ãî°“ 2. 0 «—¥ à«π‚§âß®“°®ÿ¥ (1, 0) ‰ª„π∑‘»∑“ßµ“¡‡¢Á¡π“Ãî°“ 3. = 0 ®ÿ¥ª≈“¬ à«π‚§âߧ◊Õ®ÿ¥ (1, 0) Y (x, y)
Y
O
(1, 0)
X
O
X
(x, y)
0
(1, 0)
0
‡¡◊ËÕ°”Àπ¥®”π«π®√‘ß „Àâ ®– “¡“√∂À“®ÿ¥ (x, y) ´÷Ë߇ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« || Àπ૬ „π∑‘»∑“ß°“√«—¥∑’Ë°”À𥉥â‡æ’¬ß®ÿ¥‡¥’¬«‡∑à“π—Èπ ∂â“ || 2p °“√«—¥ à«π‚§âß®–‡°‘π 1 √Õ∫ °”Àπ¥øíß°å™—π f : R Æ R ·≈– g : R Æ R ‚¥¬∑’Ë ”À√—∫·µà≈–®”π«π®√‘ß „¥Ê f() = x ·≈– g() = y ‡¡◊ËÕ (x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âߢÕß«ß°≈¡Àπ÷ËßÀπ૬∑’Ë«—¥®“°®ÿ¥ (1, 0) ¬“« || ‡√’¬°øíß°å™—π g «à“ øíß°å™—π‰´πå (sine) ·≈–‡√’¬°øíß°å™—π f «à“ øíß°å™—π‚§‰´πå (cosine) ‡¢’¬π ·∑π g ¥â«¬ sin ·≈–‡¢’¬π·∑π f ¥â«¬ cos ®–‰¥â y = sin Õà“π«à“ «“¬‡∑à“°—∫‰´πå∑µ’ “ x = cos Õà“π«à“ ‡Õ°´å‡∑à“°—∫§Õ ∑’µ“ ®“°°√“ø¢Õߧ«“¡ —¡æ—π∏å {(x, y) RR|x2y2 = 1} ®–‡ÀÁπ«à“ 1 y 1 ·≈– 1 x 1 ¥—ßπ—Èπ §à“¢Õßøíß°å™—π‰´πå·≈–øíß°å™—π‚§‰´π凪ìπ®”π«π®√‘ßµ—Èß·µà 1 ∂÷ß 1 ‡√π®å¢Õßøíß°å™—π‰´πå·≈–øíß°å™—π‚§‰´πå§◊Õ‡´µ¢Õß®”π«π®√‘ßµ—Èß·µà 1 ∂÷ß 1 ·≈–‚¥‡¡π¢Õßøíß°å™—π∑—Èß Õߧ◊Õ‡´µ¢Õß®”π«π®√‘ß
89
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
2
= 1
2
= 1
x y
®“° ®–‰¥â À√◊Õ‡¢’¬π
2
(cos ) (sin ) 2
2
cos sin = 1
‡¡◊ËÕ ‡ªìπ®”π«π®√‘ß ‡¡◊ËÕ ‡ªìπ®”π«π®√‘ß
À¡“¬‡Àµÿ 2
À¡“¬∂÷ß (cos )2 = (cos )(cos ) 2 2 cos À¡“¬∂÷ß cos ¢Õß®”π«π®√‘ß «ß°≈¡Àπ÷ËßÀπ૬¡’√—»¡’ 1 Àπ૬ ‡ âπ√Õ∫«ß¬“« 2pr À√◊Õ 2p Àπ૬ ‡¡◊ËÕ r = 1 Àπ૬ cos
2.2 §à“¢Õßøíß°å™—π‰´πå·≈–øíß°å™—π‚§‰´πå 2.2.1 §à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß∫“ß®”π«π §«“¡¬“« à«π‚§âß«—¥®“°®ÿ¥ (1, 0) ∑«π‡¢Á¡π“Ãî°“∂÷ß®ÿ¥ (0, 1) ‡∑à“°—∫
p 2
Àπ૬
§«“¡¬“« à«π‚§âß«—¥®“°®ÿ¥ (1, 0) ∑«π‡¢Á¡π“Ãî°“∂÷ß®ÿ¥ (1, 0) ‡∑à“°—∫ p Àπ૬ §«“¡¬“« à«π‚§âß«—¥®“°®ÿ¥ (1, 0) ∑«π‡¢Á¡π“Ãî°“∂÷ß®ÿ¥ (0, 1) ‡∑à“°—∫ 3p2 Àπ૬ §«“¡¬“« à«π‚§âß«—¥®“°®ÿ¥ (1, 0) ∑«π‡¢Á¡π“Ãî°“∂÷ß®ÿ¥ (1, 0) ‡∑à“°—∫ 2p Àπ૬
§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß 0, p2 , p, 32p , 2p Y (0, 1) π (1, 0)
O
Y π 2
3π 2 (0, 1) (1, 0) 2π
3π (0, 1) 2
X
(1, 0) π
O
π (0, 1) 2
2π (1, 0)
X
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
90
p 2
0
p 2
p
3p 2
2p
sin
0
1
0 ........
1 ........
0 ........
1
cos
1
0
1 ........
0 ........
1 ........
0
øíß°å™π—
3p 2
2p
0 ........
1 ........
0 ........
1 ........
0 ........
1 ........
-p
p , 3p , 5p , 7p
§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß
4
4
4
4
Y
®“°√Ÿª P(x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« p4 Àπ૬ «—¥®“°®ÿ¥ (1, 0) ·≈–‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕß à«π‚§âß AB ®–‰¥â§Õ√å¥ PA ¬“«‡∑à“°—∫§Õ√å¥ PB
B(0, 1) P(x, y)
O
X
B(1, 0)
PA = PB 2
( x1) ( y0)
2
2
2
x 2x1y
2
=
( x0) ( y1) 2
2
2
= x y 2y1
x = y
®“° ¡°“√«ß°≈¡ 1 Àπ૬
2
x y
2
= 1
2
= 1
2
=
2x x
1 2
x =
¥—ßπ—Èπ æ‘°—¥¢Õß®ÿ¥ ¿“æ –∑âÕπ¢Õß®ÿ¥ P
,
P(x y)
§◊Õ
1 2
1 1ˆ PÊ , Ë 2 2¯
À√◊Õ
P
2ˆ Ê 2 , Ë 2 2¯
à«π®ÿ¥Õ◊ËπÊ À“‰¥â®“°
91
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
Y
Y
Ê 2 2ˆ , Ë 2 2¯ p
2ˆ Ê 2 , Ë 2 2¯ p 4 (1, 0)
p 3p = 4 4
p 5p = 4 4 2ˆ Ê 2 , Ë 2 2¯ p
O
3p 4
sin
2 2
2 2
cos
2 2
7p (2 n1)p ... 4 4
, ,
5p 4
2 2
7p 2ˆ 4 Ê 2 , Ë 2 2¯ (1, 0)
X
O
X
p 4 2ˆ Ê 2 , Ë 2 2¯
5p 4
7p 4
p 4
3p 4
2 .......... 2
2 .......... 2
2 2
2 2
2 .......... 2
2 .......... 2
2 .......... 2
2 .......... 2
2 2
2 .......... 2
2 .......... 2
2 2
§à“¢Õßøíß°å™π— ‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß
7p p 2p = 4 4 Ê 2 2ˆ , 2ˆ Ê 2 Ë 2 ¯ 3p 2 , Ë 2 2¯ 4
p 4
øíß°å™π—
Ê 2 2ˆ , Ë 2 2¯
3p 5p 7p (2 n1)p ... 4 4 4 4
,
,
, ,
5p 4
7p 4
·≈– 34p , 54p ,
‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡∫«°
§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß p6 ,
5p , 7p , 11p 6 6 6
Y B(0, 1)
®“°√Ÿª P(x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§â߬“« p6 Àπ૬
P(x, y)
«—¥®“°®ÿ¥ (1, 0) A ‡ªìπ¿“æ –∑âÕπ¢Õß®ÿ¥ P ¢â“¡·°π X ®–¡’æ‘°—¥‡ªìπ (x, y)
X
O A(x, y)
à«π‚§âß PA ¬“«‡∑à“°—∫ à«π‚§âß PB ∑”„Àâ§Õ√å¥ PA ¬“«‡∑à“°—∫§Õ√å¥ PB 2
( xx) ( yy)
2 2
4y
2
=
( x0) ( y1) 2
2
= x y 2y1
2
4y 2y2 = 0 2
2
2y y1 = 0
2
2
(x y = 1)
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
92
(2y1)(y1) = 0 2y1 = 0
À√◊Õ y1 = 0 ‡π◊ËÕß®“° (x, y) Õ¬Ÿà „π§«Õ¥√—πµå∑’Ë 1 1 ¥—ßπ—Èπ y = 2 3 2
x =
·≈– æ‘°—¥¢Õß®ÿ¥ P §◊Õ p 5p p = 6 6
Ê 3 1ˆ , Ë 2 2¯
à«π®ÿ¥Õ◊ËπÊ À“‰¥â®“°¿“æ –∑âÕπ¢Õß®ÿ¥ P
Y
Y
Ê 3 1ˆ , Ë 2 2¯ 7p 6
Ê 3 1ˆ , Ë 2 2¯ p 6
Ê 3 1ˆ , Ë 2 2¯
X
(1, 0)
11p 6 Ê 3 1ˆ , Ë 2 2¯ X (1, 0) p 6 Ê 3 1ˆ , Ë 2 2¯
7p 6 3 1ˆ , 2 2¯
O
p 6
5p 6
7p 6
11p 6
p 6
sin
1 2
1 2
1 .......... 2
1 .......... 2
1 2
1 .......... 2
1 .......... 2
1 .......... 2
cos
3 2
3 .......... 2
3 .......... 2
3 2
3 .......... 2
3 .......... 2
3 2
p p = 6 Ê Ë
øíß°å™π—
Ê 3 1ˆ , Ë 2 2¯ 11p p 2p = 6 6
Ê 3 1ˆ , Ë 2 2¯ 5p 6
3 .......... 2
§à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß„π√Ÿª 2np
11p 6
O
2np
p 6
,
5p 6
2np
5p 6
,
7p 6
2np
7p 6
11p 6
·≈–
‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡
§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß p3 ,
2p 4p 5p , 3, 3 3
Y
®“°√Ÿª
B(x, y)
P(x, y)
O
A(1, 0)
X
,
P(x y)
‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“«
p 3
Àπ૬ «—¥®“°®ÿ¥ (1, 0) B ‡ªìπ¿“æ –∑âÕπ¢Õß®ÿ¥ P ¢â“¡ ·°π Y ¡’æ‘°—¥‡ªìπ (x, y) à«π‚§âß PA ¬“«‡∑à“°—∫ à«π‚§âß PB ∑”„Àâ§Õ√å¥ PA ¬“«‡∑à“°—∫§Õ√å¥ PB
93
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
(x1) ( y0)
2
2
=
( xx) ( yy)
®–‰¥â
x =
1 2
·≈–
y =
3 2
2
3ˆ Ê1 , Ë2 2 ¯
¥—ßπ—Èπ æ‘°—¥¢Õß®ÿ¥ P §◊Õ
à«πæ‘°—¥¢Õß®ÿ¥Õ◊ËπÊ À“‰¥â®“°¿“æ –∑âÕπ¢Õß®ÿ¥ P p 2p p = 3 3 Ê 1 3ˆ , Ë 2 2¯ p 4p p = 3 3 Ê 1 3ˆ , Ë 2 2¯
Y
Y Ê1 3ˆ , Ë2 2 ¯ p 3
Ê 1 3ˆ , Ë 2 2¯ 4p 3
X (1, 0) Ê1 3ˆ , Ë2 2¯ 5p p 2p = 3 3
O
p 3
2p 3
4p 3
sin
3 2
3 2
3 .......... 2
cos
1 2
1 .......... 2
øíß°å™π—
1 2
O
5p 3
5p 3
p 3
3 .......... 2
3 2
1 .......... 2
1 2
2np
‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡
°‘®°√√¡∑’Ë 2.2.1 1.
„Àâπ—°‡√’¬π‡µ‘¡µ“√“ß„Àâ ¡∫Ÿ√≥å 180
Õß»“ ‡∑à“°—∫
p
‡√‡¥’¬π
5p 3ˆ 3 Ê1 , Ë2 2 ¯
X
(1, 0) p 3 Ê1 3ˆ , Ë2 2¯
Ê 1 3ˆ , Ë 2 2 ¯ 2p 3
§à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß„π√Ÿª 2np
2p 3
3 2
3 .......... 2
3 2
1 .......... 2
1 .......... 2
1 2
p 3
,
2np
4p 3
2p 4p 2np 3 3
,
5p 3
·≈–
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
94 (
Õß»“)
(
‡√‡¥’¬π)
sin
cos
0
0
1 0 ............................... ...............................
30
p 6
1 3 ............................... ............................... 2 2
45
p 4
2 2 ............................... ............................... 2 2
60
p 3
1 3 ............................... ............................... 2 2
90
p 2
0 1 ............................... ...............................
120
2p 3
1 3 ............................... ............................... 2 2
135
3p 4
2 2 ............................... ............................... 2 2
150
5p 6
3 1 ............................... ............................... 2 2
180
p
1 0 ............................... ...............................
210
7p 6
3 1 ............................... ............................... 2 2
225
5p 4
2 2 ............................... ............................... 2 2
240
4p 3
1 3 ............................... ................................ 2 2
270
3p 2
0 1 ............................... ...............................
300
5p 3
1 3 ............................... ............................... 2 2
315
7p 4
2 2 ............................... ............................... 2 2
330
11p 6
3 1 ............................... ............................... 2 2
360
2p
1 0 ............................... ...............................
95
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2.
„Àâπ—°‡√’¬πÀ“§à“¢Õß sin ·≈– cos ‡¡◊ËÕ ‡ªìπ®”π«π®√‘ß ‡¢’¬π®”π«π®√‘ß 2np ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ ·≈– 0 2p ®”π«π®√‘ß
·π«§‘¥
sin
„π√Ÿª
cos
1) 5p
2pp
1 0 .......................... ..........................
2) 3p
(2pp) = 2pp
1 0 .......................... ..........................
9p 2 7p 4) 2
3)
4p
p 2
3p
0 1 .......................... .......................... p 3p = 2p 2 2
0 1 .......................... ..........................
5) 63p
62pp
1 0 .......................... ..........................
6) 47p
46pp
1 0 .......................... ..........................
p 3p = 12p 2 2 p 6p 2
0 1 .......................... ..........................
27p 2 13p 8) 2
7)
9) 31p 10) 67p 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
9p 4 13p 3 19p 6 21p 4 23p 6 29p 3 17p 6 100 p 3 200 p 6 95p 6
13p
0 1 .......................... ..........................
30pp
1 0 .......................... ..........................
66pp
1 0 .......................... ..........................
p 4 p 4p 3 p 7p 3p = 2p 6 6 p 5p 5p = 4p 4 4 5p 11p Ê 3p ˆ = 2p Ë 6¯ 6 2 pˆ 5p Ê 9p = 8p Ë 3¯ 3 5p 5p Ê 2 p ˆ = 2p Ë 6¯ 6 p 4p 33p = 32p 3 3 2p 4p 33p = 32p 6 3 5p 11p 15p = 14p 6 6
2 .......................... 2 3 .......................... 2 1 .......................... 2 2 .......................... 2 1 .......................... 2 3 .......................... 2 1 .......................... 2 3 .......................... 2 3 .......................... 2 1 .......................... 2
2p
2 .......................... 2 1 .......................... 2 3 .......................... 2 2 .......................... 2 3 .......................... 2 1 .......................... 2 3 .......................... 2 1 .......................... 2 1 .......................... 2 3 .......................... 2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
96 3.
°”Àπ¥«ß°≈¡Àπ÷ËßÀπ૬ „Àâπ—°‡√’¬π‡µ‘¡æ‘°—¥¢Õß®ÿ¥ª≈“¬¢Õß à«π‚§âß∫π«ß°≈¡ Y
1 3 ........) (........, 2 2 2 2 2π (........, ........) 2 2 3π 3 4 3 1 5π (........, ........) 2 2 6 1 ........) 0 (........,
3 1 (........, ........) 2 2
0 ........) 1 (........,
π 2
π 7π 6
π 3
1 3 (........, ........) 2 2 2 2 π (........, ........) 2 2 4 π 3 1 (........, ........) 6 2 2 2π
O
X 1 ........) 0 (........,
11π 5π 3 1 4 4π 7π 6 (........, ........) 2 2 2 2 (........, ........) 5π 3π 3 4 2 2 2 2 3 2 (........, ........) 1 3 2 2 (........, ........) 2 2 1 3 0 ..........) 1 (.........., ..........) (.........., 2 2
4.
„Àâπ—°‡√’¬π∫Õ°®”π«π®√‘ß∫«° 2 ®”π«π ®”π«π®√‘ß≈∫ 2 ®”π«π∑’Ë∑”„Àâ 1) sin = 1
=
2) cos = 1
3p 7p p 5p 2 2 2 2
,
,
,
3) sin = 1
4) cos = 1
3p 7p , 2 , p2 , 52p = ......................................................... 2
5) sin =
2 2
p 3p , , p4 , 34p = ......................................................... 4 4 7) sin =
p, 3p, p, 3p = .........................................................
3 2
4 p 5p , 3 , p3 , 23p = ......................................................... 3
0, 2p, 2p, 4p = ......................................................... 6) cos =
1 2
p 5p , , p3 , 53p = ......................................................... 3 3 8) cos =
1 2
4p 2p 4p 2p = ......................................................... , , , 3 3 3 3
97
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
2.2.2 §à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß„¥Ê Y
®“°®ÿ¥ (x, y) ·≈– (x, y) √ÿª‰¥â«à“ (x, y)
x = cos
O
(1, 0) (x, y)
y = sin
x = cos()
X
y = sin()
sin() = sin
¥—ßπ—Èπ
cos() = cos
∂â“ 2p ·≈–À“√ ¥â«¬ 2p ·≈⫉¥â n ‡À≈◊Õ‡»… (·Õ≈ø“) π—Ëπ§◊Õ
+
= 2np
‡¡◊ËÕ n I ·≈– 0 2p
°“√«—¥ à«π‚§âߢÕß«ß°≈¡Àπ÷ËßÀπ૬®“°®ÿ¥ (1, ‡æ√“– 2np · ¥ß«à“«—¥§√∫ n √Õ∫ √ÿª‰¥â¥—ßπ’È
0)
‰ª¬“« Àπ૬ «—¥‰ª Àπ૬°ÁæÕ
sin = sin(2np )
= sin
cos = cos(2np )
= cos
µ—«Õ¬à“ß∑’Ë 1 ®ßÀ“§à“¢Õß sin 253p , sin ÊË 256pˆ¯ , cos 254p ·≈– cos ÊË 293pˆ¯ «‘∏’∑”
sin
25p 3
p = sin Ê 8p ˆ Ë 3¯
25pˆ sin Ê Ë 6 ¯
p = sin Ê 24 p ˆ Ë 3¯
= sin =
p = sin Ê 22 p ˆ Ë 6¯
= sin
µÕ∫
= cos
25p 4
29pˆ cos Ê Ë 3 ¯
p = cos Ê 6 p ˆ Ë 4¯
=
2 2
p 6
1 2
= cos
µÕ∫ 29p 3
5p = cos Ê 8p ˆ Ë 3¯
p = cos Ê 23p ˆ Ë 4¯
= cos
25p 6
p = sin Ê 4 p ˆ Ë 6¯
p 3
3 2
= sin
p 4
= cos
µÕ∫
=
1 2
5p 3
µÕ∫
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
98
°“√À“§à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ßµ—Èß·µà 0 ∂÷ß 2p 1.
‡¡◊ËÕ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë 2
„Àâ P1(x1, y1) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ p „Àâ = p ®–‰¥â 0 2 ·≈– = p ®ÿ¥ P(x, y) ‡ªìπ¿“æ –∑âÕπ¢Õß P1 ¢â“¡·°π Y ®ÿ¥ P(x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬
Y
P1(x1, y1)
P(x, y)
O
B
A(1, 0)
X
y = sin
¥—ßπ—Èπ ·µà ·≈–
x = cos
·≈–
y = y1 = sin = sin(p ) x = x1 = cos = cos(p )
p 2 p 0 2
sin = sin(p ) = sin
‡¡◊ËÕ 0
cos = cos(p ) = cos
‡¡◊ËÕ
µ—«Õ¬à“ß∑’Ë 2 ®ßÀ“ sin 56p ·≈– cos «‘∏’∑”
sin
5p 6
2p 3
p = sin Ê p ˆ Ë 6¯
= sin =
2.
cos
p 6
p 2p = cos Ê p ˆ Ë 3¯ 3
= cos
1 2
=
‡¡◊ËÕ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë 3 ‡¡◊ËÕ p
Y P(x, y)
Q
B P1(x1, y1)
O
Ê p pˆ Ë2 ¯
A(1, 0)
X
3p 2
p 3
1 2
Ê p 3pˆ Ë 2¯
‡¢’¬π„π√Ÿª = p ‚¥¬∑’Ë 0
„Àâ P1(x1, y1) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®ÿ¥ Q ‡ªìπ°“√ –∑âÕπ¢Õß®ÿ¥ P1 ¢â“¡·°π X ®ÿ¥ P(x, y) ‡ªìπ¿“æ –∑âÕπ¢Õß®ÿ¥ Q ¢â“¡·°π Y à«π‚§âß AP ¬“« Àπ૬ y1 = y, x1 = x ®ÿ¥ P(x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬
p 2
99
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
¥—ßπ—Èπ
p 2 p 0 2
sin = sin (p ) = sin
‡¡◊ËÕ 0
cos = cos (p ) = cos
‡¡◊ËÕ
µ—«Õ¬à“ß∑’Ë 3 ®ßÀ“ sin 54p ·≈– cos 76p «‘∏’∑”
sin
p = sin Ê p ˆ Ë 4¯
5p 4
= sin =
3.
cos
p 7p = cos Ê p ˆ Ë 6¯ 6
p 4
= cos
2 2
=
µÕ∫
‡¡◊ËÕ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë 4 3p 2p 2 ‚¥¬∑’Ë 0 p2
‡¡◊ËÕ
Y P(x, y)
O
A(1, 0)
3 2
µÕ∫
Ê 3 p 2 pˆ Ë 2 ¯
‡¢’¬π„π√Ÿª = 2p
„Àâ P1(x1, y1) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®ÿ¥ P(x, y) ‡ªìπ¿“æ –∑âÕπ¢Õß®ÿ¥ P1 ¢â“¡·°π X à«π‚§âß AP ¬“« Àπ૬ ·≈– y1 = y, x1 = x ‡¡◊ËÕ P(x, y) ‡ªìπ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬
X
P1(x1, y1)
¥—ßπ—Èπ
p 6
p 2 p 0 2
sin = sin (2p ) = sin
‡¡◊ËÕ 0
cos = cos (2p ) = cos
‡¡◊ËÕ
µ—«Õ¬à“ß∑’Ë 4 ®ßÀ“§à“¢Õß sin 53p ·≈– cos 74p «‘∏’∑”
sin
5p 3
p = sin Ê 2 p ˆ Ë 3¯
= sin =
3 2
cos
p 3
p 7p = cos Ê 2 p ˆ Ë 4¯ 4
= cos
µÕ∫
=
2 2
p 4
µÕ∫
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
100
°‘®°√√¡∑’Ë 2.2.2 °”Àπ¥
1.
1) 2) 3) 4) 2.
p = 3.1416
2p = 6.2832
p 2 p 3 p 4 p 6
3p 2 2p 3 3p 4 5p 6
= 1.5708 = 1.0472 = 0.7854 = 0.5236
= 4.7124 = 2.0944 = 2.3562 = 2.6180
1 ·≈– 2 °”Àπ¥ sin = 0.48 ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®–Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë ........................ 3 ·≈– 4 °”Àπ¥ sin = 0.52 ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®–Õ¬Ÿà „π§«Õ¥√—πµå∑’Ë .................... 1 ·≈– 4 °”Àπ¥ cos = 0.91 ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®–Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë ....................... ·≈– 3 °”Àπ¥ cos = 0.85 ®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®–Õ¬Ÿà„π§«Õ¥√—πµå∑’Ë .2...................
„Àâπ—°‡√’¬π‡¢’¬π§à“¢Õßøíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ßµàÕ‰ªπ’È „ Àâ Õ ¬Ÿà „π√Ÿª§à“¢Õß øíß°å™—π‰´πå·≈–‚§‰´πå¢Õß®”π«π®√‘ß∑’Ë¡’§à“µ—Èß·µà 0 ∂÷ß p2 ‡¢’¬π„π√Ÿª ®”π«π®√‘ß
= p
cos
= 2np p 6
p sin Ê p ˆ Ë 6¯
2p 5
2p sin Ê p ˆ Ë 5¯
7p 6
p
3p 5
p
5p 3
sin
p 2 p .......................... 3
= sin
= sin
p 6
2p 5
p p sinÊ 2 p ˆ = sin ........................................... Ë 3¯ 3
p cos Ê p ˆ Ë 6¯
= cos
p 6
2p cos Ê p ˆ Ë 5¯
= cos
2p 5
p p cosÊ 2 p ˆ = cos ................................................... Ë 3¯ 3
101
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
‡¢’¬π„π√Ÿª ®”π«π®√‘ß
= p
cos
= 2np
9p 10
p p .......................... 10
p p sinÊ p ˆ = sin ........................................... Ë 10¯ 10
p p cosÊ p ˆ = cos ................................................... Ë 10¯ 10
17p 9
p 2 p .......................... 9
p p sinÊ 2 p ˆ = sin ........................................... Ë ¯ 9 9
p p cosÊ 2 p ˆ = cos ................................................... Ë ¯ 9 9
15p 7
p 2 p .......................... 7
p p sinÊ 2 p ˆ = sin ........................................... Ë 7¯ 7
p p cosÊ 2 p ˆ = cos ................................................... Ë 7¯ 7
19p 5
p Ê 4 p ˆ .......................... Ë 5¯
p p sinÊ 4 p ˆ = sin ........................................... Ë 5¯ 5
p p cosÊ 4 p ˆ = cos ................................................... Ë 5¯ 5
7p 3
p .......................... Ê 2 p ˆ Ë 3¯
p p sinÊ 2 p ˆ = sin ........................................... Ë 3¯ 3
p p cosÊ 2 p ˆ = cos ................................................... Ë 3¯ 3
3.
sin
°”Àπ¥„Àâ 0
p 2
·≈– sin = 0.42 ®ßÀ“§à“¢Õß
1) cos
2) sin(p)
«‘∏’∑” ®“°
2
2
sin cos = 1
«‘∏’∑”
2
2
cos = 1sin
‡π◊ËÕß®“° 0 cos =
sin(p) = sin
p 2
0.42 = .............................. 2
1sin
=
2
1(0.42) 1 0.1764 = ..............................
0.8236 = .............................. = 0.91 .............................. 3) cos (p)
«‘∏’∑”
cos (p)
4) cos (2p)
«‘∏’∑”
cos (2p)
= cos
= cos (2p)
= 0.91 ..............................
) = cos(2p ..............................
[
]
= cos .............................. = 0.91 ..............................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
102 5) sin (2p)
«‘∏’∑”
4.
6) cos (p)
sin (2p)
«‘∏’∑”
cos(p) ...................................
sin = ..............................
cos = ..............................
0.42 = ..............................
0.91 = ..............................
∂â“ cos2xsin2x =
1 2
®ßÀ“§à“ x ‡¡◊ËÕ 0 x p 2
2
cos xsin x =
«‘∏’∑” ®“° 2
2
cos x(1cos x) =
1 2 1 2
2
1 .............................. 2
2 cos x =
2
3 .............................. 2
2
3 .............................. 4
2
cos x1cos x =
cos x =
3 ..............................
cos x =
‡¡◊ËÕ
p 0 x 2 cos x =
2
‡¡◊ËÕ 3 2
p x p 2 cos x =
p
cos x =
cos .................... 6
x =
.................... 6
p
p 5p ¥—ßπ—Èπ §à“¢Õß x §◊Õ .......... ·≈– .......... 6 6
3 2
pˆ Ê cos x = cos Ë p 6 ¯
....................
x =
p p .................... 6
x =
5p .................... 6
103
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 5.
®ßÀ“‡´µ§”µÕ∫¢ÕßÕ ¡°“√ 1) sin cos , 0 p
2) sin cos p 2p
,
Y
Y
π 4 O
(1, 0)
X
5π 4
È p , p˘ ‡´µ§”µÕ∫§◊Õ............................................. ÍÎ 4 ˙˚
O
(1, 0)
È5p , 2p˘ ‡´µ§”µÕ∫§◊Õ............................................. ÍÎ 4 ˙˚
2.3 øíß°å™—πµ√’ ‚°≥¡‘µ‘Õ◊ËπÊ ∫∑𑬓¡ ”À√—∫®”π«π®√‘ß „¥Ê sin tan = ‡¡◊ÕË cos cot = cosec = sec =
cos sin 1 sin 1 cos
cos π 0
‡¡◊ÕË
sin π 0
‡¡◊ÕË
sin π 0
‡¡◊ÕË
cos π 0
‚¥‡¡π·≈–‡√π®å¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘ 1. ‚¥‡¡π¢Õßøíß°å™—π sin ·≈– cos §◊Õ R 2. ‡√π®å¢Õßøíß°å™—π sin ·≈– cos §◊Õ {x R|1 x 1} (2 n1)p , 3. ‚¥‡¡π¢Õßøíß°å™—π tan ·≈– sec §◊Õ R ÏÌx R | x = 2 Ó ‚¥‡¡π¢Õßøíß°å™—π cot ·≈– cosec §◊Õ R{x R|x = np, n I} 5. ‡√π®å¢Õßøíß°å™—π tan ·≈– cot §◊Õ R 6. ‡√π®å¢Õßøíß°å™—π sec ·≈– cosec §◊Õ R{x R|1 x 1} 4.
X
n I ¸˝ ˛
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
104
§«“¡ —¡æ—π∏å√–À«à“ßøíß°å™—πµ√’ ‚°≥¡‘µ‘µà“ßÊ 1 cot = ‡¡◊ÕË tan π 0 À√◊Õ sin π 0 tan 2
sec
2
2
cosec
1tan =
2
1cot =
‡¡◊ÕË ‡¡◊ÕË
cos
π 0
sin
π 0
°‘®°√√¡∑’Ë 2.3 1.
°”Àπ¥ sin = 0.52 ·≈– 0 2
p 2
®ßÀ“§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘Õ◊ËπÊ ¢Õß
2
1) sin cos = 1 2
2
cos = 1sin
p 2
2
cos =
1sin , 0 2
1(0.52) = ............................................................ 10.2704 = ............................................................ 0.7296 = ............................................................
0.85 = ............................................................
2.
sin cos 0.52 = .............................. 0.85 0.61 = ..............................
2)
tan =
4)
sec =
1 cos 1 = .............................. 0.85 1.18 = ..............................
°”Àπ¥„Àâ 0 1) cosec sec 2
2
3) sec tan
«‘∏∑’ ”
p 2
·≈– sin =
3 5
1 sin 1 = .............................. 0.52 1.92 = ..............................
3)
cosec =
5)
cot =
cos sin 0.85 = .............................. 0.52 1.63 = ..............................
®ßÀ“§à“¢Õß 2
2
2) sin cos 2
2
4) cosec cot
2 ®“° cos = 1sin ‡π◊ËÕß®“° 0 p2 ¥—ßπ—Èπ cos = 1sin 2
2
3 1Ê ˆ cos = ........................................ Ë 5¯
105
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 9 1 = ........................................ 25
®–‰¥â
16 = ........................................ 25 4 = ........................................ 5 3 4 tan = .................... cot = .................... 4 3 5 5 sec = .................... cosec = .................... 4 3 2
1) cosec sec
2 2 Ê 3ˆ Ê 4ˆ = ........................................ Ë 5¯ Ë 5¯
5 5 = ........................................ 3 4 2015 = ........................................ 12 35 = ........................................ 12 2
9 16 = ........................................ 25 25
= 1........................................
2
2
3) sec tan 2
3.
2
2) sin cos
2
4) cosec cot
2
Ê 5ˆ Ê 3ˆ = ........................................ Ë 4¯ Ë 4¯
2 2 Ê 5ˆ Ê 4ˆ = ........................................ Ë 3¯ Ë 3¯
25 9 = ........................................ 16 16
25 16 = ........................................ 9 9
1 = ........................................
= 1........................................
„Àâπ—°‡√’¬π‡µ‘¡µ“√“ß· ¥ß§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘ ‡¡◊ËÕ 0
sin
cos
tan
p 2
cosec
sec
cot
‰¡à𑬓¡
1 ....................
‰¡à𑬓¡
0
0 0 1 .................... .................... ....................
p 6
1 3 3 2 3 3 2 .................... .................... .................... .................... .................... .................... 2 2 3 3
p 4
2 2 2 2 .................... .................... .................... .................... .................... .................... 1 1 2 2
p 3
3 2 3 3 1 3 2 .................... .................... .................... .................... .................... .................... 3 3 2 2
p 2
0 1 .................... ....................
‰¡à𑬓¡
.................... 1
‰¡à𑬓¡
.................... 0
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
106 4.
„Àâπ—°‡√’¬π∫Õ°«à“®ÿ¥ª≈“¬ à«π‚§âß∑’ˬ“« Àπ૬ ®–Õ¬Ÿà„π§«Õ¥√—πµå„¥ ‡¡◊ËÕ°”Àπ¥ 1 2 1) sin 0 ·≈– cos 0 ............... 2) sin 0 ·≈– cos 0 ............... 4 3 3) sin 0 ·≈– cos 0 ............... 4) sin 0 ·≈– cos 0 ............... 3 2 6) sin 0 ·≈– tan 0 ............... 5) tan 0 ·≈– cos 0 ............... 2 4 7) cos 0 ·≈– tan 0 ............... 8) cos 0 ·≈– tan 0 ...............
5.
„Àâπ—°‡√’¬πÀ“§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘∑ÿ°øíß°å™—π¢Õß®”π«πµàÕ‰ªπ’È
®ÿ¥ª≈“¬ à«π‚§âßÕ¬Ÿà„π §«Õ¥√—πµå∑’Ë
1)
p 2p Ê p ˆ 3¯ 3 Ë
2 .......................
3 1 1 2 2 ............ 3 ............ ............ ............ ............ ............ 2 2 3 3
2)
p 3p Ê p ˆ 4¯ 4 Ë
2 .......................
2 2 2 2 1 ............ ............ 1 ............ ............ ............ ............ 2 2 2 2
3)
p 5p Ê p ˆ Ë 6¯ 6
2 .......................
1 2 1 3 2 ............ 3 ............ ............ ............ ............ 3 3 ............ 2 2
4)
p 7p Ê p ˆ 6¯ 6 Ë
3 .......................
2 1 1 3 2 ............ 3 ............ ............ ............ ............ 3 ............ 3 2 2
5)
p 5p Ê p ˆ 4¯ 4 Ë
3 .......................
2 2 2 2 1 1 ............ ............ 2 ............ 2 ............ 2 2 ............ ............
6)
p 4p Ê p ˆ Ë 3¯ 3
3 .......................
2 1 1 3 3 ............ 2 ............ ............ ............ ............ ............ 3 3 2 2
7)
p 5p Ê 2p ˆ Ë 3¯ 3
4 .......................
3 1 2 1 2 3 ............ ............ ............ 2 ............ 2 3 ............ ............ 3
8)
p 7p Ê 2 p ˆ Ë 4¯ 4
4 .......................
2 2 2 2 1 ............ 1 ............ ............ 2 ............ 2 2 ............ 2 ............
9)
p 11p Ê 2 p ˆ 6¯ 6 Ë
4 .......................
3 1 1 2 2 ............ ............ 3 ............ ............ ............ ............ 2 2 3 3
sin
cos
tan cosec sec
cot
107
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
®ÿ¥ª≈“¬ à«π‚§âßÕ¬Ÿà„π §«Õ¥√—πµå∑’Ë
p 5p Ê p ˆ Ë 4¯ 4
2 .......................
2 2 2 2 1 ............ ............ 1 ............ ............ ............ ............ 2 2 2 2
p 13p 11) Ê 3p ˆ Ë 4¯ 4
2 .......................
2 ............ ............ 2 ............ 2 ............ 2 ............ ............ 1 1 2 2 2 2
p 25p Ê 4 p ˆ Ë 6¯ 6
4 .......................
3 1 1 2 2 ............ ............ 3 ............ ............ ............ ............ 2 2 3 3
10)
12)
6.
sin
cos
tan cosec sec
cot
®ßÀ“§à“¢Õß 14 pˆ 1) sin Ê Ë 3 ¯
‡π◊ËÕß®“°æÀÿ§Ÿ≥¢Õß 2p §◊Õ 2p, 4p ·≈– 6p ·≈– 6p 143p 4p 14 p 3
‡¢’¬π„π√Ÿª
18p 4 p 4p = 6p 3 3 3
14 pˆ 4p sin Ê = sin È 3(2 p)˘ ÍÎ 3 ˙˚ Ë 3 ¯
¥—ßπ—πÈ
4p = sin 3
..............................
= 3 2
..............................
13pˆ 2) cos Ê Ë 3 ¯
‡π◊ËÕß®“°æÀÿ§Ÿ≥¢Õß 2p §◊Õ 2p, 4p ·≈– 6p ·≈– 6p 133p 4p 13p 3
‡¢’¬π„π√Ÿª
6p
5p 3
13pˆ 5p cos Ê = cos È 3(2 p)˘ ÍÎ 3 ˙˚ Ë 3 ¯
=
5p cos .............................. 3
=
1 .............................. 2
14 p p = 5p 3 3 14 p 2p = 4p 3 3
®ÿ¥ª≈“¬ à«π‚§âßÕ¬Ÿà∑’˧«Õ¥√—πµå∑’Ë 3
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
108 7.
„Àâπ—°‡√’¬πÀ“§à“¢Õß 1) sin
5p 7p 3p 4p tan cos sin 6 6 4 3
p p p p = sin Ê p ˆ tan Ê p ˆ cos Ê p ˆ sin Ê p ˆ Ë Ë Ë Ë 6¯ 6¯ 3¯ 4¯ p p p p = sin tan Êcos ˆ Êsin ˆ Ë ¯ Ë 3¯ 4 6 6
1 3 2 3 Ê ˆ Ê ˆ = ..................................................................................... 2 3 Ë 2 ¯Ë 2 ¯ 1 3 6 = ..................................................................................... 2 3 4 64 33 6 = ..................................................................................... 12 2) cos
25p 15p 16 p 13p cos sin sin 4 4 3 4
p p p p .... .... .... .... = cos Ê 6p 4 ˆ cos Ê 4 p 4 ˆ sin Ê 5p 3 ˆ sin Ê 3p 4 ˆ Ë ¯ Ë ¯ Ë ¯ Ë .... .... .... .... ¯ p p p p cos cos Ê sin ˆ Ê sin ˆ = ..................................................................................... 3¯ Ë 4¯ 4 4 Ë
3ˆ Ê 2ˆ Ê 2 ˆ Ê 2 ˆÊ = ..................................................................................... Ë 2¯Ë 2¯ Ë 2¯Ë 2¯ 2 6 = ..................................................................................... 4 4 2 6 = ..................................................................................... 4 3) sin
5p 2p 7p p tan cos cot Ê ˆ Ë 3¯ 3 6 6
p ˆ p ˆ .... p p Ê .... Ê = sin Á p 3 ˜ tan Á p 6 ˜ cos cot Ê 2 pˆ Ë3 ¯ .... ¯ Ë .... ¯ Ë 6
p p p p sin tan cos cot = ..................................................................................... 3 6 6 3 Ê 3 ˆ Ê 1 ˆÊ 3 ˆ Ê 1 ˆ = ..................................................................................... Ë 2 ¯ Ë 3¯ Ë 2 ¯ Ë 3¯
1 1 = ..................................................................................... 2 2 1 = .....................................................................................
p 5p p 2 6 7p 3p p 6 2 p 3p p 2 4 4p 3p p 3 2
109
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
4) sin
9p 11p 29p cos sin cos(5p) 4 3 6
p p p sinÊ 2 p ˆ cosÊ 4 p ˆ sinÊ 5 p ˆ cos 5 p = ..................................................................................... Ë Ë Ë 3¯ 6¯ 4¯
p p p sin cos sin cos 5 p = ..................................................................................... 4 3 6 Ê 2 ˆ Ê 1 ˆÊ 1 ˆ (1) = ..................................................................................... Ë 2 ¯ Ë 2¯ Ë 2¯ 2 1 = ..................................................................................... 4 2 22 = ..................................................................................... 4 2
5) 3tan
p 4 2p 1 2p 1 2p cos sec sin 6 3 6 2 4 3 3 2
2
2
Ê 1 ˆ 4 Ê 3ˆ 1 ( )2 1 Ê 3ˆ 3 2 = ..................................................................................... Ë 3¯ 3 Ë 2 ¯ 2 3Ë 2 ¯
1 111 = ..................................................................................... 4 1 1 = ..................................................................................... 4 3 = ..................................................................................... 4 6) sin
p p 2p 2p cos Ê ˆ sin cos Ê ˆ Ë 6¯ Ë 6¯ 3 3
Ê 3ˆ Ê 3ˆ Ê 3ˆ Ê 3ˆ = ..................................................................................... Ë 2 ¯Ë 2 ¯ Ë 2 ¯Ë 2 ¯ 3 3 = ..................................................................................... 4 4
0 = ..................................................................................... 8.
∂â“ x ‡ªìπ®”π«π®√‘ß∫«°∑’Ë¡’§à“πâÕ¬°«à“ 2p ·≈â« 2cos 2x ¡’§à“¡“°∑’Ë ÿ¥ ‡¡◊ËÕ x ¡’§à“‡∑à“‰√ «‘∏∑’ ” ®“°°”Àπ¥ 0 x 2p ·≈– 2cos 2x ¡’§à“¡“°∑’Ë ÿ¥ ∂â“ 2cos 2x ¡’§à“¡“°∑’Ë ÿ¥·≈â« cos 2x µâÕß¡’§à“πâÕ¬∑’Ë ÿ¥ §à“πâÕ¬∑’Ë ÿ¥¢Õß cos 2x §◊Õ 1 ¥—ßπ—πÈ cos 2x = 1
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
110
cos 2x = cos(2npp)
[cos p = 1]
2x = 2npp
p np x = ......................................................................... 2
‡¡◊ËÕ n = 0,
p x = ......................................................................... 2
‡¡◊ËÕ n = 1,
p 3p p = x = ......................................................................... 2 2
‡¡◊ËÕ n = 2,
p 5p 2p = 2p x = ......................................................................... 2 2
´÷Ëß¡“°°«à“
p 3p ·≈– ¥—ßπ—Èπ x ¡’§à“‡∑à“°—∫ ................................. 2 2 9.
∂â“ x ‡ªìπ®”π«π®√‘ß∫«°¡’§à“πâÕ¬°«à“ 2p ·≈â« 3sin 3x ¡’§à“πâÕ¬∑’Ë ÿ¥ ‡¡◊ËÕ x ¡’§à“‡∑à“‰√ «‘∏∑’ ” ®“°°”Àπ¥ 0 x 2p ·≈– 3sin 3x ¡’§à“πâÕ¬∑’Ë ÿ¥ µâÕß¡’§à“¡“°∑’Ë ÿ¥ ∂â“ 3sin 3x ¡’§à“πâÕ¬∑’Ë ÿ¥·≈â« sin 3x ................................................................. §à“¡“°∑’Ë ÿ¥¢Õß sin 3x §◊Õ 1 sin 3x = 1......................................................................... p sin 3x = sin Ê 2np ˆ ........................................................................................................................................ Ë 2¯
¥—ßπ—πÈ
p 3x = 2np ........................................................................................................................................ 2 2np p x = ........................................................................................................................................ 3 6 p n = 0 x = ......................................................................... 6
‡¡◊ËÕ
,
‡¡◊ËÕ n = 1,
2p p 5p = x = ......................................................................... 3 6 6
‡¡◊ËÕ n = 2,
4p p 3p = x = ......................................................................... 3 6 2
p 2p 2p x = ......................................................................... 6 p 5p 3p x ........................................................................................................................................ 6 6 2
‡¡◊ËÕ n = 3,
¥—ßπ—Èπ ¡’§à“‡∑à“°—∫ ,
´÷Ëß¡“°°«à“
·≈–
111
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
2.4 øíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß¡ÿ¡ Àπ૬„π°“√«—¥¡ÿ¡ §◊Õ Õß»“ 1 Õß»“ ‡∑à“°—∫ 60 ≈‘ª¥“ () 1 ≈‘ª¥“ ‡∑à“°—∫ 60 øî≈‘ª¥“ () ¡ÿ¡∑’Ë®ÿ¥»Ÿπ¬å°≈“ߢÕß«ß°≈¡´÷Ëß√Õß√—∫¥â«¬ à«π‚§âߢÕß«ß°≈¡∑’ˬ“«‡∑à“°—∫√—»¡’¢Õß «ß°≈¡¡’¢π“¥ 1 ‡√‡¥’¬π a
¡ÿ¡∑’®Ë ¥ÿ »Ÿπ¬å°≈“ߢÕß«ß°≈¡√—»¡’ r Àπ૬ ´÷ßË √Õß√—∫¥â«¬ à«π‚§âߢÕß«ß°≈¡∑’¬Ë “« a Àπ૬ ®–¡’¢π“¥ a ‡√‡¥’¬π r
r
Õß»“
=
‡√‡¥’¬π
=
1 1
®“°√Ÿª ®–‰¥â 180 Õß»“ p 180 180 p
‡√‡¥’¬π Õß»“
⬇
0.01745
=
a r
=
p
‡√‡¥’¬π
‡√‡¥’¬π
⬇ 57 18
à«π‚§âߢÕß«ß°≈¡Àπ÷ßË Àπ૬∑’√Ë Õß√—∫¡ÿ¡∑’®Ë ¥ÿ »Ÿπ¬å°≈“ß¢π“¥ 1 ‡√‡¥’¬π ®–¬“« 1 Àπ૬ ¥—ßπ—Èπ à«π‚§âߢÕß«ß°≈¡Àπ÷ËßÀπ૬∑’Ë√Õß√—∫¡ÿ¡∑’Ë®ÿ¥»Ÿπ¬å°≈“ß¢π“¥ ‡√‡¥’¬π ®÷߬“« Àπ૬ Ÿ °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ´÷Ëß¡’ C ‡ªìπ¡ÿ¡©“° ¥—ß√Ÿª B
®“°√Ÿª √ÿª‰¥â¥—ßπ’È =
a c
,
sin B
=
b c
cos A =
b c
,
cos B =
a c
tan A
=
a b
,
tan B
=
b a
cot A
=
b a
,
cot B
=
a b
sin A c
A
a
b
C
129
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
°‘®°√√¡∑’Ë 2.5 1.
2.
3.
„Àâπ—°‡√’¬πÕà“π§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘®“°µ“√“ß 1) sin 24 40
0.4173 = ................................
2) cos 72
0.9090 = ................................
3) tan 55 20
1.4460 = ................................
4) cot 41 50
1.1171 = ................................
5) sin 68 20
0.9283 = ................................
6) cos 35 30
0.8141 = ................................
7) tan 39 50
0.8342 = ................................
8) cot 65 20
2.1775 = ................................
9) sin 44 50
0.7050 = ................................
10) cos 45 10
0.7050 = ................................
11) tan 42 40
0.9217 = ................................
12) cot 47 20
0.9217 = ................................
„Àâπ—°‡√’¬πÕà“π§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß®“°µ“√“ß 1) sin 0.3142
0.3090 = ................................
2) cos 1.2566
0.3092 = ................................
3) tan 0.3985
0.4210 = ................................
4) cot 1.1723
0.4210 = ................................
5) sin 0.9657
0.8225 = ................................
6) cos 0.6050
0.8225 = ................................
7) tan 1.2392
2.9042 = ................................
8) cot 0.3316
2.9042 = ................................
9) sin 0.8319
0.7392 = ................................
10) cos 0.7389
0.7392 = ................................
11) tan 1.5417
34.368 = ................................
12) cot 0.0271
34.368 = ................................
„Àâπ—°‡√’¬πÀ“§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π®√‘ß 1) sin 0.495
®“°µ“√“ß
sin 0.4945 = 0.4746 sin 0.4974 = 0.4772
§à“¢Õß®”π«π®√‘ßµà“ß°—π 0.0029 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π §à“¢Õß®”π«π®√‘ßµà“ß°—π (0.4950.4945) = 0.0005 0005 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π 00..0029 (0.0026) ª 0.0004 ¥—ßπ—πÈ
0.0026
0.47460.0004 sin 0.495 = ............................................................ 0.475 = ............................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
130
(2) sin 0.8862 0.7735 ........................................................................ 0.7753 sin 0.8872 = ........................................................................ 0.0018 0.0029 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π .................. §à“¢Õß®”π«π®√‘ßµà“ß°—π .................. .88620.8843) = ........................................................................ 0.0019 §à“¢Õß®”π«π®√‘ßµà“ß°—π (0............................ 0.0019 (0.0018) ª ........................................................................ 0.0012 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π ............................ 0.0029 0.77350.0012 ¥—ßπ—Èπ sin 0.8862 = ........................................................................ 0.7747 = ........................................................................
®“°µ“√“ß
sin 0.8843 =
(3) cos 1 cos 0.9977 = 0 5422
. ........................................................................ 0.5398 cos 1.0007 = ........................................................................ 0.0024 0.003 §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π .................. §à“¢Õß®”π«π®√‘ßµà“ß°—π .................. (10.9977) = ........................................................................ 0.0023 §à“¢Õß®”π«π®√‘ßµà“ß°—π ............................ 0.0023 (0.0024) ª ........................................................................ 0.0018 §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π............................ 0.003 0.54220.0018 ¥—ßπ—Èπ cos 1 = ........................................................................ 0.5404 = ........................................................................
®“°µ“√“ß
(4) cos 0.33 cos 0.3287 = 0 9465
. ........................................................................ 0.9455 cos 0.3316 = ........................................................................ 0.0029 §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π .................. 0.001 §à“¢Õß®”π«π®√‘ßµà“ß°—π .................. (0.330.3287) = ........................................................................ 0.0013 §à“¢Õß®”π«π®√‘ßµà“ß°—π ............................ 0.0013 0.0004 (0.001) ª ........................................................................ §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π............................ 0.0029 0.94650.0004 ¥—ßπ—Èπ cos 0.33 = ........................................................................ 0.9461 = ........................................................................
®“°µ“√“ß
(5) tan 0.5575
. ................................................................... 0.6249 tan 0.5585 = ................................................................... 0.0029 §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π .................. 0.0041 §à“¢Õß®”π«π®√‘ßµà“ß°—π .................. (0............................ .55750.5556) = ................................................................... 0.0019 §à“¢Õß®”π«π®√‘ßµà“ß°—π 0.0019 0.0027 (0.0041) ª ................................................................... §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π ............................ 0.0029 ®“°µ“√“ß
tan 0.5556 = 0 6208
131
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 ¥—ßπ—Èπ
tan 0.5575 = 0 62080 0027 =
. . ................................................................... 0.6235 ...................................................................
6) tan 1.5 tan 1.4981 = 13 727
. ................................................................... 14.301 tan 1.5010 = ................................................................... 0.0029 §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π .................. 0.574 §à“¢Õß®”π«π®√‘ßµà“ß°—π .................. (1.51.4981) = ................................................................... 0.0015 §à“¢Õß®”π«π®√‘ßµà“ß°—π ............................ 0.0019 0.376 (0.574) ª ................................................................... §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π ............................ 0.0029 13.7270.376 ¥—ßπ—Èπ tan 1.5 = ................................................................... 14.103 = ...................................................................
®“°µ“√“ß
4.
„Àâπ—°‡√’¬πÀ“¢π“¥¢Õß¡ÿ¡ 1) sin A
=
A
®“°∑’Ë°”Àπ¥„Àâ
0.4234 0.4226 = sin 25 0
®“°µ“√“ß
0.4253 = sin 25 10
§à“¢Õßøíß°å™—π‰´πåµà“ß°—π 0.0027 ¢π“¥¢Õß¡ÿ¡µà“ß°—π 10 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π (0.42340.4226) = 0.0008 0008 ¢π“¥¢Õß¡ÿ¡µà“ß°—π 00..0027 (10) ⬇ 3 A = 25 03
¥—ßπ—πÈ
25 3 = ................................................................................. 2) sin A
=
®“°µ“√“ß
0.6826 0.6820 = sin 43 0 ..................................
0.6841 = sin 43 10 ........................................................ 10 0.0021 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π ............................. ¢π“¥¢Õß¡ÿ¡µà“ß°—π ................................ 0.68260.6820 = 0.0006 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π .................................................................................................... 0.0006 (10) ª 3 ¢π“¥¢Õß¡ÿ¡µà“ß°—π ............................................................................................................. 0.0021 43 0 3 ¥—ßπ—πÈ A = .............................................................................. 43 3 = .................................................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
132
3) sin A
=
0.4790 sin 28 30 0.4772 = ................................................................................
®“°µ“√“ß
sin 28 40 0.4797 = ................................................................................ 10 0.0025 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π ............................. ¢π“¥¢Õß¡ÿ¡µà“ß°—π ................................ 0.47900.4772 = 0.0018 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π .................................................................................................... 0.0018 (10) ª 7 ¢π“¥¢Õß¡ÿ¡µà“ß°—π ............................................................................................................. 0.0025 28 30 7 ¥—ßπ—πÈ A = ................................................................................. 28 37 = ................................................................................. 4) cos A
=
®“°µ“√“ß
0.9412 0.9417 = cos 19 40 ................................ 0.9407 = cos 19 50 ................................
§à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π 0.001 ¢π“¥¢Õß¡ÿ¡µà“ß°—π 10 §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π 0.94170.9412 = 0.0005 ¢π“¥¢Õß¡ÿ¡µà“ß°—π 00..0005 (10) ⬇ 5 0001 ¥—ßπ—πÈ A = 19 405 45 = 19 ................................................................................ 5.
„Àâπ—°‡√’¬πÀ“§à“¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß¡ÿ¡ 1) sin 24 43
®“°µ“√“ß
sin 24 40 = 0.4173 sin 24 50 = 0.4200
¢π“¥¢Õß¡ÿ¡µà“ß°—π ¢π“¥¢Õß¡ÿ¡µà“ß°—π
10
¥—ßπ—πÈ
sin
3
§à“¢Õßøíß°å™—π‰´πåµà“ß°—π 0.0027 §à“¢Õßøíß°å™—π‰´πåµà“ß°—π 103 (0.0027) = 0.0008 0.41730.0008 24 43 = ................................................................................. 0.4181 = .................................................................................
2) cos 64 26
®“°µ“√“ß
cos 64 20 = 0.4331 cos 64 30 = 0.4305
¢π“¥¢Õß¡ÿ¡µà“ß°—π ¢π“¥¢Õß¡ÿ¡µà“ß°—π
10 6
§à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π §à“¢Õßøíß°å™—π‚§‰´πåµà“ß°—π
0.0026
6 (0.0026) 10
=
0.0016
133
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 cos 64 26 = 0.43310.0016
¥—ßπ—πÈ
0.4315 = ................................................................................. 3) tan 77 12 4.3892 tan 77 10 = .................................................................................
®“°µ“√“ß
4.4494 tan 77 20 = ................................................................................. 10 §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π 0.0597 ¢π“¥¢Õß¡ÿ¡µà“ß°—π .......... ......................................... 2 2 (0.0597) = 0.0119 ¢π“¥¢Õß¡ÿ¡µà“ß°—π .......... §à“¢Õßøíß°å™—π·∑π‡®πµåµà“ß°—π 10 .................. .................. .38970.0119 ¥—ßπ—πÈ tan 77 12 = 4................................................................................. .4016 = 4................................................................................. 4) cot 40 36
. 1.1640 cot 40 40 = ................................................................................. 0.0068 10 §à“¢Õßøíß°å™—π‚§·∑π‡®πµåµà“ß°—π ..................................... ¢π“¥¢Õß¡ÿ¡µà“ß°—π .......... 1 1708 cot 40 30 = .................................................................................
®“°µ“√“ß
6
(0.0068) = 0.0041 6 ¢π“¥¢Õß¡ÿ¡µà“ß°—π .......... §à“¢Õßøíß°å™—π‚§·∑π‡®πµåµà“ß°—π 10 ................. .............. .17080.0041 ¥—ßπ—πÈ cot 40 36 = 1................................................................................. .1667 = 1................................................................................. 6.
°”Àπ¥„Àâ 1)
0 x 180
„Àâπ—°‡√’¬πÀ“
sin x = 0.7526
x
‚¥¬‡ªî¥µ“√“ß 2)
sin x = sin 48.8
sin x = sin 72.1 x = 72.1 ...................................................................
x = 48.8
sin 72 1 = sin(180 72 1 ) ...................................................................
sin 48.8 = sin(180 48.8 )
¥—ßπ—πÈ 3)
sin x = 0.9515
.
.
= sin 131.2
= sin 107 9 ...................................................................
x = 48.8 131.2
¥—................................................................... ßπ—πÈ x = 72.1 , 107.9
.
,
cos x = 0.2375 = cos 76.3
4)
cos x = 0.7826 = cos 38.5
= cos (180 76.3 )
= cos(180 38 5 ) ...................................................................
= cos 103.7
= cos 141 5 ...................................................................
x = 103.7
.
.
x = 141 5 ...................................................................
.
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
134 tan x = 0.7814
5)
tan x = 1.215
6)
= tan 38
= tan 50.5
= tan(180 38 )
= tan(180 50 5 ) ...................................................................
= tan 142
= tan 129 5 ...................................................................
.
.
x = 129 5 ...................................................................
.
x = 142 7.
0 360
°”Àπ¥„Àâ
sin =
1)
„Àâπ—°‡√’¬πÀ“§à“
2 2
®“° ¡°“√ cos = cos 180 .................
sin = sin 45 = sin(180 45 )
¥—ßπ—Èπ
8.
= 180 ...................................................................
45 , 135 = ........................................
sin = 0.6180
3)
cos = 1
2)
cos = 0.5125
4)
sin = sin 38 10 ...................................................................
cos = cos 59 10 ...................................................................
= sin(180 38 10 ) ...................................................................
= 59 10 ...................................................................
= 218 10 ...................................................................
cos = cos(360 59 10 ) ...................................................................
sin = sin(360 38 10 ) ...................................................................
= 300 50 ...................................................................
= 321 50 ...................................................................
...................................................................
„À⇵‘¡‡§√◊ËÕßÀ¡“¬ ¡“°°«à“
()
πâÕ¬°«à“
()
À√◊Õ‡∑à“°—∫
(=)
≈ß„π™àÕß«à“ß„Àâ∂Ÿ°µâÕß
1) sin 55 ................. cos 55
= 2) sin 50 ................. cos 40
3) sin 36 ................. cos 18
4) sin
5) sin
3p 2
................. cos 0
= 7) sin 54 ................. cos 36 9) tan 15 ................. cot 15 11) tan
2p 3
p ................. cot 3
p 2 p 6) sin 5
................. cos p
4p ................. cos 5 = 8) sin 18 ................. cos 72 10) tan
3p 4
12) tan p
3p ................. cot 4 p ................. sin 2
135
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
9.
°”Àπ¥„Àâ
sin
p 4
=
0.7
·≈–
cos
p 4
=
1) 2cos 45 3sin 135
0.7
„Àâπ—°‡√’¬πÀ“§à“¢Õß
2) 3cos 135 4sin 135
= 2 cos 45 3 sin 45 ...................................................................
= 3(cos 45 )4 sin 45 ...................................................................
= 2(0 7)3(0 7) ...................................................................
= 3(0 7)4(0 7) ...................................................................
= 35 ...................................................................
= 07 ...................................................................
.
.
.
3) cos 135 2sin 45
.
.
.
4) 2cos 315 3sin 45
= cos 45 2 sin 45 ...................................................................
= 2 cos 45 3 sin 45 ...................................................................
= 0 72(0 7) ...................................................................
= 2(0 7)3(0 7) ...................................................................
.
.
= 2.1 ...................................................................
5p 7p sin 4 4 p p = 2Ê cos ˆÊ sin ˆ ................................................................... Ë ¯ Ë 4 4¯
5) 2cos
. . = 0.7 ................................................................... 7p 5p 3sin 4 4 p p Ê = 4 cos 3 sin ˆ ................................................................... Ë 4¯ 4
6) 4cos
= 2(0 7)(0 7) ...................................................................
= 4(0 7)3(0 7) ...................................................................
= 2 1 ...................................................................
= 49 ...................................................................
.
.
.
11p 9p 4cos 4 4 p p = 3 sin 4 cos ................................................................... 4 4
7) 3sin
.
.
.
13p 21p cos 4 4 p p = 5Ê sin ˆÊ cos ˆ ................................................................... Ë ¯ Ë 4 4¯
8) 5sin
= 3(0 7)4(0 7) ...................................................................
= 5(0 7)(0 7) ...................................................................
= 49 ...................................................................
= 2 8 ...................................................................
.
.
.
5p 3p 9) sin Ê ˆ cos Ê ˆ Ë 4¯ Ë 4¯
.
.
.
7p 3p 10) cos Ê ˆ 2sin Ê ˆ Ë 4¯ Ë 4¯
5p 3p = sin cos ................................................................... 4 4
3p 7p = cos 2Ê sin ˆ ................................................................... Ë 4¯ 4
p p = Ê sin ˆÊ cos ˆ ................................................................... Ë 4¯ Ë 4¯
7p 3p = cos 2 sin ................................................................... 4 4 p p = cos 2 sin ................................................................... 4 4 = 0 72(0 7) ...................................................................
= (0 7)(0 7) ...................................................................
.
.
= 0 ................................................................... ...................................................................
. . = 2.1 ...................................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
136
10.
°”Àπ¥„Àâ sin 2
0 2p
=
cos
«‘∏∑’ ” ®“° ‡π◊ÕË ß®“°
„Àâπ°— ‡√’¬πÀ“®”π«π®√‘ß∫«°
´÷ßË Õ¥§≈âÕß°—∫ ¡°“√
1 2 1 sin 2 = cos 2
1 p 1 cos = sin Ê ˆ Ë2 2 ¯ 2 p 1 sin 2 = sin Ê ˆ Ë2 2 ¯
2 =
¥—ßπ—πÈ
p 1 2 2
p 1 2 = 2p Ê ˆ Ë2 2 ¯ p 1 2 = 4p Ê ˆ Ë2 2 ¯ p 1 2 = 6p Ê ˆ Ë2 2 ¯
¥—ßπ—πÈ
‡π◊ÕË ß®“°
1 2 = 2
cos
5 2 5 2 5 2 5 2
= = = =
p 2 5p 2 9p 2 13p 2
®–‰¥â
=
p ............................................. 5
®–‰¥â
=
p .............................................
®–‰¥â
=
®–‰¥â
=
9p ............................................. 5 13 p ............................................. 5
1 p 1 = sin Ê ˆ Ë 2 2 ¯ 2
p 1 sin 2 = sin Ê ˆ Ë2 2 ¯
¥—ßπ—πÈ
p 1 2 = ................................................................................ 2 2 p 1 2 pÊ ˆ 2 = ................................................................................ Ë2 2 ¯ p 1 4 pÊ ˆ 2 = ................................................................................ Ë2 2 ¯ p 1 6 pÊ ˆ 2 = ................................................................................ Ë2 2 ¯
137
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
¥—ßπ—πÈ
1 2 2
§”µÕ∫∑’˵âÕß°“√§◊Õ 11.
p p ®–‰¥â = ............................................. 3 2 5 p 5p = ®–‰¥â = ............................................. 3 2 9 p 9p = ®–‰¥â = ............................................. 3 2 13 p 13p = ®–‰¥â = ............................................. 3 2 p , p , 95p , p3 , 53p ......................................................................................................... 5 =
Ÿ
3 2 3 2 3 2 3 2
Ÿ
√Ÿª “¡‡À≈’ˬ¡ ABC ¡’ A = 30 , C Ÿ ¢Õß B ·≈–§«“¡¬“«¢Õߥâ“π∑’ˇÀ≈◊Õ «‘∏∑’ ”
=
= 90
·≈–
a = 9 3
‡´πµ‘‡¡µ√ ®ßÀ“¢π“¥ B
c
a= 9 3
30 A
Ÿ
°”Àπ¥„Àâ
A Ÿ
C
30 = ............................................. 90 = .............................................
Ÿ
¥—ßπ—πÈ sin A
C
b
=
a c
À√◊Õ
60 B = ............................................. 9 3 sin 30 = ............................................. c
1 2
9 3 = ............................................. c
c = ............................................. 29 3 ............ c = ............................................. 18 3 ............ b cos 30 = ............................................. 18 3
‡´πµ‘‡¡µ√
cos A
=
b c
À√◊Õ
3 2
b = ............................................. 18 3
3 18 3 b = ............................................. 2 b = ............................................. 27 ............
‡´πµ‘‡¡µ√
´¡.
138
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
12.
√Ÿª “¡‡À≈’ˬ¡ ABC ¡’¥â“π BC ¬“« 25 2 ‡´πµ‘‡¡µ√ ¥â“π AB ¬“« 50 ‡´πµ‘‡¡µ√ ¡ÿ¡ C ¡’¢π“¥ 90 ®ßÀ“¢π“¥¢Õß¡ÿ¡ B §«“¡¬“«¢Õߥâ“π AC ·≈–‡ âπµ—Èß©“°∑’Ë≈“°®“° C ‰ª¬—ߥâ“π AB «‘∏∑’ ” C
25 2
A
45
45 D 50
°”Àπ¥
´¡.
B
´¡.
‡´πµ‘‡¡µ√ 50 ‡´πµ‘‡ ¡µ√ .............................................
25 2 BC = ............................................. AB = Ÿ
90 = ............................................. BC cos B = AB 25 2 = ............................................. 50 C
2 = ............................................. 2 Ÿ
45 ¥—ßπ—πÈ B = ............................................. Ÿ 45 ·≈– A = ............................................. “¡‡À≈’ˬ¡Àπâ“®—Ë« √Ÿª “¡‡À≈’ˬ¡ ABC ‡ªìπ√Ÿª ............................................. 25 2 ‡´πµ‘‡¡µ√ ¥—ßπ—πÈ AC = ............................................. „π√Ÿª “¡‡À≈’ˬ¡ BCD; sin B = CD BC CD sin 45 = ............................................. 25 2 1 CD ....................................................... = ............................................. 2 25 2
‡´πµ‘‡¡µ√
25 CD = .............................................
139
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
13.
®“°√Ÿª “¡‡À≈’ˬ¡ ABC À“§«“¡¬“«¢Õß AD ·≈– «‘∏∑’ ”
Ÿ
B = 45
,
Ÿ
C = 120 , a
= 40
‡´πµ‘‡¡µ√ „Àâπ—°‡√’¬π
A
CD
45 B 40
Ÿ
°”Àπ¥
ABC Ÿ
BCA Ÿ
¥—ßπ—πÈ
D
´¡. C
ACD
120
45 = ............................................. 120 = ............................................. 60 = .............................................
40 BC = .............................................
‡´πµ‘‡¡µ√
„π√Ÿª “¡‡À≈’ˬ¡
ACD
; Ÿ
tan ACD
=
tan 60 = 3 =
AD =
„π√Ÿª “¡‡À≈’ˬ¡
ABD
AD CD AD CD AD CD 3 CD
..........(1)
; Ÿ
AD BD AD tan 45 = 40 CD 3CD 1 = ............................................................................................................................................. 40CD tan ABD =
3CD 40+CD = ............................................................................................................................................. 40 = ( 31) CD ............................................................................................................................................. 40 Ê 31ˆ CD = Á ˜ ............................................................................................................................................. 31 Ë 31¯ 40( 31) ............................................................................................................................................. = 31 = 20 ( 31) .............................................................................................................................................
‡´πµ‘‡¡µ√
(1) AD = 3 [20( 3 1)] .............................................................................................................................................
®“° ;
= 20(3 3 ) .............................................................................................................................................
‡´πµ‘‡¡µ√
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
140
14.
Ÿ
®“°√Ÿª∑’Ë°”Àπ¥ Q ®ßÀ“§«“¡¬“«¢Õß «‘∏∑’ ”
= 90 QR = 8
,
RS
,
QS
·≈–
‡´πµ‘‡¡µ√,
Ÿ
Ÿ
QRP = 60 QPR = 30
,
∂â“
QS ⬜ PR
PS Q
8
R
„π√Ÿª “¡‡À≈’ˬ¡
´¡. 60
30 S
P
;
QRS
RS RQ RS 1 = ............................................................................ 8 2 1 8 RS = ............................................................................ 2
cos 60 =
‡´πµ‘‡¡µ√
RS = 4............................................................................ QS RS QS 3 = ............................................................................ 4
tan 60 =
4 3 QS = ............................................................................
‡´πµ‘‡¡µ√
„π√Ÿª “¡‡À≈’ˬ¡
PQS
; QS PS 4 3 = ............................................................................ PS
tan 30 = 1 3
4 3 3 PS = ............................................................................
‡´πµ‘‡¡µ√
12 PS = ............................................................................
141
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
15.
√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π ABCD ¥â“π AB ¬“« 24 27 3 ‡´πµ‘‡¡µ√ ¥â“π AD ¬“« 14 Ÿ ‡´πµ‘‡¡µ√ BAD = 30 , DE ⬜ AB ∑’Ë®ÿ¥ E ®ßÀ“§«“¡¬“«¢Õß DE , AE ·≈– BD D 14
A
C
´¡.
30
247 3
«‘∏∑’ ” „π√Ÿª “¡‡À≈’ˬ¡
B
E
´¡.
ADE sin 30 =
;
DE DA
1 DE = ............................................................................................................................................. 2 DA 1 DE = DA ............................................................................................................................................. 2 1 = 14 ............................................................................................................................................. 2 = 7 .............................................................................................................................................
‡´πµ‘‡¡µ√
AE AD AE 3 ............................................................................................................................................. = AD 2 cos 30 =
3 AE = AD ............................................................................................................................................. 2 3 = 14 ............................................................................................................................................. 2 = 7 3 .............................................................................................................................................
‡´πµ‘‡¡µ√
„π√Ÿª “¡‡À≈’ˬ¡
BDE
;
BD
2
2
= DE BE 2
2
2
= 7 (24) ............................................................................................................................................. = 49576 ............................................................................................................................................. = 625 ............................................................................................................................................. BD = 25 .............................................................................................................................................
¥—ßπ—πÈ
‡´πµ‘‡¡µ√
142
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
16.
µ÷° ÕßÀ≈—ß Ÿßµà“ß°—π 30 ‡¡µ√ ™“¬§πÀπ÷Ë߬◊πÕ¬Ÿà∑’Ëæ◊Èπ·π«√“∫‡¥’¬«°—∫µ÷° Àà“ß®“°µ÷°∑’ˇµ’Ȭ °«à“‡ªìπ√–¬– 100 ‡¡µ√ —߇°µ‡ÀÁπ¬Õ¥µ÷°Õ¬Ÿà„π·π«‡ âπµ√߇¥’¬«°—π‡ªìπ¡ÿ¡‡Õ’¬ß 27 2 °—∫æ◊Èπ√“∫ ®ßÀ“§«“¡ Ÿß¢Õßµ÷°∑—Èß Õß (°”Àπ¥ tan 27 2 = 0.51) D 30 B
BC DE Ÿ
E
C 100
A
F
27 2
A
«‘∏∑’ ” „Àâ
¡.
¡.
‡ªìπ®ÿ¥ —߇°µ ‡ªì𧫓¡ Ÿß¢Õßµ÷°∑’ˇµ’Ȭ°«à“ ‡ªì𧫓¡ Ÿß¢Õßµ÷°∑’Ë Ÿß°«à“
DAE =
27 2
„π√Ÿª “¡‡À≈’ˬ¡
;
ABC
BC AC
tan 27 2 =
BC = AC tan 27 2 1000.51 = .............................................
¥—ßπ—È𠧫“¡ Ÿß¢Õßµ÷°À≈—ß∑’ˇµ’Ȭ°«à“
51 = .............................................
‡¡µ√
5130 DE = .............................................
¥—ßπ—È𠧫“¡ Ÿß¢Õßµ÷°À≈—ß∑’Ë Ÿß°«à“ 17.
‡¡µ√
81 = .............................................
√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π ABCD ¡’‡ âπ√Õ∫√Ÿª¬“« 12 ‡´πµ‘‡¡µ√ DE µ—Èß©“°°—∫ AB ∑’Ë E DE = 3 ‡´πµ‘‡¡µ√ ¡ÿ¡ A ¡’¢π“¥ 60 Õß»“ ®ßÀ“æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π√Ÿªπ’È D
C
3
´¡.
60 A
E
B
143
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
«‘∏’∑” √Ÿª “¡‡À≈’ˬ¡
ADE
‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“° sin A =
DE AD
sin 60 =
3 AD
3 3 AD = ............................................................................ o = 3 sin 60 2
‡´πµ‘‡¡µ√
2 = ............................................................................ 12 ABBCCDDA = ............................................................................ 12 2(AB)2(DA) = ............................................................................ ABDA = ............................................................................ 6 ...................................................... 62 = 4 AB = ............................................................................
‡´πµ‘‡¡µ√
æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡
ABDE ABCD = ............................................................................ 4 3 = ............................................................................
µ“√“߇´πµ‘‡¡µ√
4 3 = ............................................................................ 18.
√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ABC ¡’ AD µ—Èß©“°°—∫ BC ∑’Ë®ÿ¥ D „Àâ §«“¡¬“«¢Õß AD ·≈–®ß· ¥ß«à“ cos260 cot230 = 134
BC = 2x
Àπ૬ ®ßÀ“
A
B
«‘∏∑’ ” ®“°√Ÿª
C
D
Àπ૬ 2x Àπ૬ ............................................................................
x BD = ............................................................................ AB = 2
AD
2
2
2
(2x) x = 3x = ............................................................................
3x AD = ............................................................................ Ÿ
ABD Ÿ
BAD
60 = ............................................................................ 30 = ............................................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
144 Ÿ
BD x 1 = = = ...................................................... AB 2x 2
Ÿ
3x AD = = 3 = ....................................................... x BD
cos ABD = cos 60 cot BAD = cot 30 2
Ê 1 ˆ ( 3 )2 2 2 cos 60 cot 30 = ............................................................................ Ë 2¯
¥—ßπ—πÈ
1 3 = ............................................................................ 4 13 = ............................................................................ 4
19.
≥ ∑’Ë®ÿ¥Àπ÷Ëß¡Õ߇ÀÁπ‡ “∏߇ªìπ¡ÿ¡‡ß¬¢÷Èπ 30 ‡¡◊ËÕ‡¥‘π‡¢â“‰ª¬—߇ “∏ßÕ’° 10 ‡¡µ√ ¡ÿ¡‡ß¬ ¢÷Èπ¢Õ߬ե‡ “∏߇ªìπ 45 ®ßÀ“§«“¡ Ÿß¢Õ߇ “∏ß „Àâ A ‡ªìπ®ÿ¥ —߇°µ§√—Èß·√° D „Àâ B ‡ªìπ®ÿ¥ —߇°µ§√—Èß∑’Ë Õß AB = 10 ‡¡µ√ „π√Ÿª “¡‡À≈’ˬ¡ ACD; A
45
30
C
B
CD AC CD 1 = ........................................... ABBC 3
tan 30 =
‡π◊ËÕß®“° BCD ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë« CD ¥—ßπ—Èπ ¥â“π BC ¬“«‡∑à“°—∫¥â“π................................................................................................................. 1 CD = ............................................................................................................................................................ 3 10CD 10CD = 3CD ............................................................................................................................................................ 3CD CD = 10 ............................................................................................................................................................ ( 31)CD = 10 ............................................................................................................................................................
10 Ê 31ˆ CD = Á ˜ ............................................................................................................................................................ 31 Ë 31¯ = 5( 31) ............................................................................................................................................................ 5( 31) ............................................................................................................................................................
¥—ßπ—Èπ ‡ “∏ß Ÿß
20.
‡¡µ√
≥ ∑’Ë®ÿ¥Àπ÷Ëß´÷ËßÀà“ß®“°™—Èπ≈à“ߢÕßµ÷°À≈—ßÀπ÷Ëß„π·π«√“∫ 40 ‡¡µ√ ¡Õ߇ÀÁπ¬Õ¥µ÷°·≈– ª≈“¬‡ “Õ“°“»´÷Ëßµ—ÈßÕ¬Ÿà∫π¬Õ¥µ÷°‡ªìπ¡ÿ¡ 30 ·≈– 60 µ“¡≈”¥—∫ ®ßÀ“«à“‡ “Õ“°“» Ÿß ‡∑à“‰√
145
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 „Àâ A ‡ªìπ®ÿ¥ —߇°µ Àà“ß®“°µ÷°„π·π«√“∫∑’Ë®ÿ¥ ¥—ßπ—Èπ AB = 40 ‡¡µ√ BC ‡ªì𧫓¡ Ÿß¢Õßµ÷° CD ‡ªì𧫓¡ Ÿß¢Õ߇ “Õ“°“» „π√Ÿª “¡‡À≈’ˬ¡ ABC;
D
B
BC tan 30 = ............................................ AB 1 BC = ............................................ 3 60 30 40 A B 40 ¡. 40 BC = ............................................ 3 ABD BD tan 60 = ............................................ AB BCCD 3 = ............................................................................................................................................................ 40 40 40 3 = CD ............................................................................................................................................................ 3 C
„π√Ÿª “¡‡À≈’ˬ¡
;
120 = 40 3CD ............................................................................................................................................................ 80 80 3 CD = = ............................................................................................................................................................ 3 3 80 3 ............................................................................................................................................................ 3
¥—ßπ—Èπ ‡ “Õ“°“» Ÿß
21.
‡¡µ√
„π°“√ ”√«®§«“¡°«â“ߢÕß·¡àπÈ” ¬◊π ”√«®§√—Èß·√°∑’Ë®ÿ¥ A ´÷ËßÕ¬Ÿàµ√ߢⓡ°—∫∑à“πÈ” ‡¡◊ËÕ ‡¥‘π∫π∂ππ‡≈’¬∫·¡àπÈ”‰ª∑’Ë®ÿ¥ B ¡Õ߇ÀÁπ∑à“πÈ”‡ªìπ¡ÿ¡ 60 ·≈–‡¥‘πµàÕ‰ªÕ’° 40 ‡¡µ√ ∑’Ë ®ÿ¥ C ¡Õ߇ÀÁπ∑à“πÈ”‡ªìπ¡ÿ¡ 45 ®ßÀ“§«“¡°«â“ߢÕß·¡àπÈ” ∑à“πÈ” D
C
45 40
60
¡.
B
A
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
146
„Àâ
D
‡ªìπ®ÿ¥·∑π∑à“πÈ”
„π√Ÿª “¡‡À≈’ˬ¡ ABD;
AD tan 60 = ...................................................................................... AB AD 3 = ...................................................................................... AB AD AB = ...................................................................................... 3
„π√Ÿª “¡‡À≈’ˬ¡ ACD ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë« AD AC = ...................................................................................... ABBC = AD ............................................................................................................................................................ AD 40 = AD ............................................................................................................................................................ 3
3AD AD 40 3 = ............................................................................................................................................................ 40 3 = ( 31)AD ............................................................................................................................................................ 40 3( 31) AD = ............................................................................................................................................................ 2
AD = 20 3( 31) ............................................................................................................................................................ 20 3( 31) ............................................................................................................................................................
¥—ßπ—Èπ ·¡àπÈ”°«â“ß
22.
‡¡µ√
‡ “‰øøÑ“ 2 µâπ Ÿß‡∑à“°—π Õ¬ŸàÀà“ß°—π 100 ‡¡µ√ ™“¬§πÀπ÷Ë߬◊πÕ¬Ÿà∫πæ◊Èπ√“∫√–À«à“ß ‡ “‰øøÑ“∑—Èß Õß ≥ ®ÿ¥∑’ˬ◊π¡Õ߇ÀÁπ‡ “‰øøÑ“ 2 µâ𠇪ìπ¡ÿ¡‡ß¬ 30 ·≈– 60 ®ßÀ“«à“ ™“¬§ππ’Ȭ◊πÕ¬Ÿà ≥ ®ÿ¥„¥ ·≈–‡ “‰øøÑ“∑—Èß Õß Ÿß‡∑à“‰√ A
C
30 B
«‘∏∑’ ” „Àâ
x
¡.
60 E 100x
¡.
·≈– CD ·∑π‡ “‰øøÑ“ 2 µâπ ´÷Ëß Ÿß‡∑à“°—π E ‡ªìπ®ÿ¥ —߇°µ„ÀâÀà“ß®“°‡ “‰øøÑ“ AB x ‡¡µ√ ¥—ßπ—πÈ BE = x ‡¡µ√ ·≈– DE = 100x ‡¡µ√ AB
tan 30 =
AB BE
D
147
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 BE tan 30 AB = ...................................................... tan 60 =
..........(1)
CD DE
DE tan 60 CD = ......................................................
..........(2)
‡π◊ÕË ß®“° AB = CD BE tan 30 = DE tan 60 ¥—ßπ—Èπ ................................................................................................................................ 1 ............................................................................................................................................. (x) Ê ˆ = (100x) 3 Ë 3¯ x = 3(100x) ............................................................................................................................................. x = 3003x ............................................................................................................................................. 4x = 300 ............................................................................................................................................. x = 75 .............................................................................................................................................
¥—ßπ—Èπ ‡¢“¬◊π ≥ ®ÿ¥´÷ËßÀà“ß®“°‡ “‰øøÑ“ µâπ ‡¡µ√ ·≈– ‡¡µ√ ‡ “‰øøÑ“ Ÿß DE tan 60 = 25 3 ‡¡µ√ ............................................................................................................................................. 2 75 25 .............................................................................................................................................
2.6 °√“ø¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘ °√“ø¢Õß
y = sin x Y
2p
3p 2
p
p 2
3p 2
1 O 1
p 2
X p
2p
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
148
°√“ø¢Õß
y = cos x
Y
1
3p 2
°√“ø¢Õß
p 2
p
p 2
O 1
3p 2
p
X
y = tan x Y
°√“ø¢Õß
3p 5p 2p p 2 2
Op 2
p 2
3p 2
p
2p
X
5p 2
y = cot x Y
2p
°√“ø¢Õß
3p p p 2 2
O
p 2
p
3p 2
2p
5p 2
2p
5p 2
X 3p
y = cosec x Y
3p 2p p 2
p 1 2
1
3p 2
O p 2
p
X 3p
149
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 y = sec x
°√“ø¢Õß
Y
1 O p 2 1
3p 5p 2p p 2 2
p 2
3p 2
p
2p
X
5p 2
‚¥‡¡π·≈–‡√π®å¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘
øíß°å™π—
‚¥‡¡π
sine
R
cosine
R
tangent
(2 k 1) p Ïxx k I ¸˝ Ì | R xπ 2 ˛ Ó
cotangent secant cosecant
‡√π®å {y|y {y|y
,
{x|x
,
R x π kp k I
,
,
,
{x|x
,
R x π kp k I
,
,
}
1
, |y| R, |y|
} 1}
R
}
Ïx|x R x π (2 k 1) p k I ¸ Ì ˝ 2 Ó ˛
R
R
{y|y
R
, |y|
1
}
{y|y
R
, |y|
1
}
§“∫·≈–·Õ¡æ≈‘®Ÿ¥ øíß°å™—πµ√’ ‚°≥¡‘µ‘∑ÿ°øíß°å™—π‡ªìπøíß°å™—π∑’ˇªìπ§“∫ ‡¡◊ËÕ·∫àß·°π X ÕÕ°‡ªìπ™à«ß¬àÕ¬ ‚¥¬∑’˧«“¡¬“«¢Õß·µà≈–™à«ß¬àÕ¬‡∑à“°—π ·≈–°√“ø„π·µà≈–™à«ß¬àÕ¬¡’≈—°…≥–‡À¡◊Õπ°—𠧫“¡¬“«¢Õß™à«ß¬àÕ¬∑’Ë —Èπ∑’Ë ÿ¥∑’Ë¡’ ¡∫—µ‘¥—ß°≈à“« ‡√’¬°«à“ §“∫ (period) ¢Õßøíß°å™—π
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
150
”À√—∫øíß°å™—π∑’ˇªìπ§“∫´÷Ëß¡’§à“ Ÿß ÿ¥‡∑à“°—∫ 1 (amplitude) ‡ªìπ (ab) 2
a
·≈–§à“µË” ÿ¥‡∑à“°—∫
{(x, y)|y = A sin Bx} {(x, y)|y = A cos Bx} ·µà≈–øíß°å™—π¡’·Õ¡æ≈‘®Ÿ¥‡ªìπ |A| ·≈–¡’§“∫‡ªìπ
b
®–¡’·Õ¡æ≈‘®Ÿ¥
øíß°å™—π
§“∫æ◊Èπ∞“π¢Õßøíß°å™—π §“∫æ◊Èπ∞“π¢Õßøíß°å™—π
,
, ·≈–
..........(1) ..........(2) 2p | B|
·≈– cosecant ‡ªìπ 2p cotangent ‡ªìπ p
sine cosine secant tangent
°‘®°√√¡∑’Ë 2.6 1.
®ß∫Õ°§“∫·≈–·Õ¡æ≈‘®Ÿ¥æ√âÕ¡∑—È߇¢’¬π°√“ø ¡°“√
§“∫
·Õ¡æ≈‘®Ÿ¥
1) y = 4sinx
2p ...........
4 ....................
°√“ø Y
4 π
1 2) y = 3sin x 2
4p ...........
π 2
O 4
π 2
π
π
2π
X
Y
3 .................... 3
2π
π
O 3
X
151
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
¡°“√
§“∫
·Õ¡æ≈‘®Ÿ¥
3) y = sin3x
2p 3
1
°√“ø Y 1
π π 3 6
O
π 6
π 3
X
π 2
π
X
π
2π
1
4) y = 2cosx
2p ...........
Y
2 .................... 2 π 2 π
O 2
1 5) y = 3cos x 2
4p ...........
Y
3 .................... 3
2π
π
O
X
3
Y
6) y = 4sinx
2p ...........
4 ....................
4
O
4
π 2
π
3π 2
2π
X
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
152
¡°“√
§“∫
·Õ¡æ≈‘®Ÿ¥
7) y = 3sin3x
2p ........... 3
3 ....................
°√“ø Y 3
O
π 6
π 3
π 2
2π 3
π
2π
3π
4π
π
2π
3π
4π
3π 4
π
X
3
1 8) y = 3sin x 2
4p ...........
3 ....................
Y 3
O
X
3
1 9) y = 2cos x 2
4p ...........
2 ....................
Y
2 O
X
2
10) y = 4sin2x
p ...........
4 ....................
Y 4
O
4
π 4
π 2
X
153
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
¡°“√ 11) y =
1 3 sin x 2 2
§“∫
·Õ¡æ≈‘®Ÿ¥
4p ........... 3
1 .................... 2
°√“ø Y 1 2 O 1 2
12) y = 2 cos
2x 3
3p ...........
π
4π 3
3π 3π 9π 4 2 4
3π
π 3
2π 3
X
Y
2 .................... 2
O
X
2
13) y =
3 1 sin x 2 4
8p ...........
3 .................... 2
Y 3 2 O
2π 4π
6π
8π
4π 2π 3
8π 3
X
3 2
14) y = 0.5 sin
3x 8 p ........... 3 4
Y
0.5 .................... 3 2
O 3 2
2π 3
X
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
154
2.
®ßÀ“·Õ¡æ≈‘®Ÿ¥·≈–§“∫æ◊Èπ∞“π æ√âÕ¡∑—È߇¢’¬π°√“ø„π™à«ß ¡°“√ 1) y = 4 sin x
·Õ¡æ≈‘®Ÿ¥ |4|
= 4
2p x 2p
§“∫
°√“ø Y
2p 4
2p 2p
p
O
X
p
4
2) y =
1 cos x 2
1 ............... 2
Y
2p ....................
1 2
2π
π
O
π
2π
π
2π
X
1 2
3) y = 3 cos x
3 ...............
Y
2p .................... 3
2π
π
O 3
X
155
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
¡°“√ 4) y =
3 sin x 2
·Õ¡æ≈‘®Ÿ¥
§“∫
3 ............... 2
2p ....................
°√“ø Y 3 2
2π π
π
O
2π
X
3 2
5) y =
4 cos x 3
4 ............... 3
2p ....................
Y 4 3
2π
π
π
O
2π
X
4 3
6) y =
5 sin x 2
5 ............... 2
2p ....................
Y 5 2
2π π
O
π
2π
X
5 2
3.
®ß‡¢’¬π°√“ø¢Õßøíß°å™—π 1) 2)
Ï( x y)|y = 2 sin 1 x ¸ Ì ˝ 3 ˛ Ó
,
·≈–‡µ‘¡§”µÕ∫≈ß„π™àÕß«à“ß
2 ·Õ¡æ≈‘®Ÿ¥‡∑à“°—∫ ........................... ·≈–§“∫æ◊Èπ∞“π‡∑à“°—∫ ‡¢’¬π°√“ø¢Õßøíß°å™—π„π™à«ß§“∫æ◊Èπ∞“π‰¥â¥—ßπ’È
6p ...........................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
156
Y 2 1 O
2π
4π
X
6π
1 2
3)
À“®ÿ¥ Ÿß ÿ¥ ‡¡◊ËÕ
3p x = ............................................. 2
¥—ßπ—Èπ
1 3p 2 sin Ê ˆ = 2 y = ............................................. 3Ë 2 ¯
®ÿ¥ Ÿß ÿ¥§◊Õ 4)
Ê 3 p , 2ˆ ..................................................................... Ë 2 ¯ 9p x = ............................................. 2
À“®ÿ¥µË” ÿ¥ ‡¡◊ËÕ
¥—ßπ—Èπ
1 9p y = ............................................. 2 sin Ê ˆ 3Ë 2 ¯
= ............................................. 2
®ÿ¥µË” ÿ¥§◊Õ
Ê 9 p , 2ˆ .................................................................... Ë 2 ¯
2.7 øíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õߺ≈∫«°·≈– º≈µà“ߢÕß®”π«π®√‘ßÀ√◊Õ¡ÿ¡ ‡¡◊ËÕ , ‡ªìπ®”π«π®√‘ßÀ√◊Õ¡ÿ¡„¥Ê cos ( ) = cos cos sin sin
.....(1)
cos ( ) = cos cos sin sin
.....(2)
sin ( ) = sin cos cos sin
.....(3)
sin ( ) = sin cos cos sin
.....(4)
tan ( ) =
tan tan 1 tan tan
,
cot ( ) =
cot cot 1 cot cot
tan ( ) =
tan tan 1 tan tan
,
cot ( ) =
cot cot 1 cot cot
157
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 µ—«Õ¬à“ß∑’Ë 1 ®ßÀ“§à“¢Õß cos 15 , cos 75 , sin 15 ·≈– sin 75 «‘∏’∑” cos 15 cos 75 = cos (45 30 ) À√◊Õ cos (60 45 ) = cos (30 45 ) = cos 45 cos 30 sin 45 sin 30
= cos 30 cos 45 sin 30 sin 45
À√◊Õ cos 60 cos 45 sin 60 sin 45
=
3 2 1 2 2 2 2 2
=
2 3 2 1 2 2 2 2
=
6 2 4 4
=
6 2 4 4
=
6 2 4
=
6 2 4
µÕ∫
sin 15
µÕ∫ sin 75
= sin (60 45 )
À√◊Õ sin (45 30 )
= sin 60 cos 45 cos 60 sin 45
= sin (30 45 ) = sin 30 cos 45 cos 30 sin 45
=
3 2 1 2 2 2 2 2
=
=
6 2 4 4
=
6 2 4 4
=
6 2 4
=
6 2 4
√ÿª
cos 15 = sin 75
µÕ∫ ·≈– cos 75 = sin 15
µ—«Õ¬à“ß∑’Ë 2 ®ßÀ“§à“¢Õß tan 15 ·≈– tan 75 «‘∏’∑” tan 15 = tan (60 45 ) À√◊Õ tan (45 30 ) =
=
= =
o o tan 60 tan 45 o o 1 tan 60 tan 45
31 1 31 31 31 31 31
3 2 3 1 31
1 2 3 2 2 2 2 2
tan 75 = tan (30 45 ) =
o o tan 30 tan 45 o o 1 tan 30 tan 45
1 1 3 = 1 1 1 3
= =
31 31 31 31
3 2 3 1 31
µÕ∫
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
158
=
42 3 2
=
= 2 3
µÕ∫
42 3 2
= 2 3
µÕ∫
®“° (1), (2), (3) ·≈– (4) ®–‰¥â 2 sin cos = sin ( )sin ( )
.....(5)
2 cos sin = sin ( )sin ( )
.....(6)
2 cos cos = cos ( )cos ( )
.....(7)
2 sin sin = cos ( )cos ( )
.....(8)
= x
∂â“
= y xy 2
=
®–‰¥â
·≈– =
xy 2
®“° (5), (6), (7) ·≈– (8) ®–‰¥â xy xy cos 2 2 xy xy sin xsin y = 2 cos sin 2 2 xy xy cos xcos y = 2 cos cos 2 2 xy xy cos xcos y = 2 sin sin 2 2
sin xsin y = 2 sin
øíß°å™—πµ√’ ‚°≥¡‘µ‘¢Õß®”π«π´÷Ë߇ªìπ Õ߇∑à“ “¡‡∑à“¢Õß sin 2 = 2 sin cos 2
2
cos 2 = cos sin 2
cos 2 = 2 cos 1 2
cos 2 = 12 sin tan 2 =
2 tan 2 1 tan
cot 2 =
cot 1 2 cot
2
159
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
cos 2 = sin 2 =
1 tan 2
1 tan
2 tan 2 1 tan 3
sin 3 = 3 sin 4 tan 3
cos 3 = 4 cos 3 cos 3
tan 3 =
®“° cos 2x ®–‰¥â cos x 2x
cos cos
13 tan 2
2x
= 2 cos
1
®–‰¥â
2 1 cos x 2
x 2
= 1 cos x 2
=
=
1 cos x 2 1 cos x 2
µ—«Õ¬à“ß∑’Ë 3 ®ßÀ“ tan
p 8
=
x 2
= 1 cos x 2
x tan 2
x x 2 cos 2 2 x x cos 2 cos 2 2
sin
sin x cos x
x 2 x cos 2
=
2
1 cos x 2
2
sin
sin
x 2
2x
cos x = 12 sin 2x
=
®“°
2
cos 2x = 12 sin x
·≈–
2
x tax 2
«‘∏’∑”
2
= 2cos x1
tan x =
tan
3 tan tan
À√◊Õ
=
=
1 cos x 1 cos x
tan
sin
tan
x 2
p tan 8
=
sin x 1 cos x
p = tan 4 2
=
sin
p 4
1 cos
p 4
x 2
x x cos 2 2 2x 2 cos 2
2 sin
=
sin x 1 cos x
=
sin x 1 cos x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
160
2 2
=
1
2 2
2 2 2 2 2 2 2
=
=
2 (2 2) 42
=
2 1
°‘®°√√¡∑’Ë 2.7 1.
1) sin 2 = 2 sin cos
®ß· ¥ß«à“
2
2
2
2
2) cos 2 = cos sin = 2 cos 1 = 12 sin 3) tan 2 =
«‘∏’∑”
1)
2 tan 2 1 tan
sin 2 = sin ( ) cos sin = sin cos ................................................................................. 2 sin cos = .........................................................................................................
2)
cos 2 = cos ( ) sin sin = cos cos ................................................................................. 2
2
sin = cos ......................................................................................................... 2
2
À√◊Õ (1sin2 )sin2 2 2 2 2 1sin sin cos 1cos ..................................................... = ..................................................... 2 2 12 sin 2 cos 1 ..................................................... = .....................................................
= cos (1cos ) = = 3)
tan 2 = tan ( )
tan tan 1 tan tan 2 tan = ......................................................................................................... 2 1 tan =
161
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2.
3
1) sin 3 = 3 sin 4 sin
®ß· ¥ß«à“
3
2) cos 3 = 4 cos 3 cos 3
3) tan 3 =
«‘∏’∑”
1)
3 tan tan 2
13 tan
sin 3 = sin (2 ) = sin 2 cos cos 2 sin 2
= (2 sin cos ) cos (12 sin ) sin 2
3
sin cos sin 2 sin = 2........................................................................................................ 2
3
sin (1sin )sin 2 sin = 2........................................................................................................ 3
3
sin 2 sin sin 2 sin = 2........................................................................................................ 3
sin 4 sin = 3........................................................................................................ 2)
cos 3 = cos (2 ) = cos 2 cos sin 2 sin 2
= (2 cos 1) cos (2 sin cos ) sin 3
2
2 cos cos 2 cos sin = ........................................................................................................ 2
2
2
3
2 cos cos 2 cos (1cos ) = ........................................................................................................ 2 cos cos 2 cos 2 cos = ........................................................................................................ 3
4 cos 3 cos = ........................................................................................................ 3)
tan 3 = tan (2 ) tan 2 tan = ........................................................................................................ 1 tan 2 tan 2 tan tan 2 = ........................................................................................................ 1 tan 2 tan 1 tan 2 1 tan 3
2 tan tan tan = ........................................................................................................ 2 2 1 tan 2 tan 3
3 tan tan = ........................................................................................................ 2 1 3 tan
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
162
3.
∂â“ sin x = 14 ·≈– p x
3p 2
®ßÀ“§à“¢Õß cos 2x ·≈– sin 2x
2
2
cos xsin x = 1
«‘∏’∑” À“ cos x ®“°
2
2
cos x = 1sin x
‡π◊ËÕß®“° p x
3p 2
®–‰¥â cos x =
2
1 sin x ..................................................................... 2
·∑π§à“ ®–‰¥â
1 Ê 1ˆ cos x = ..................................................................... Ë 4¯ 15 = ..................................................................... 4
„™â µŸ √ sin 2 = 2sin cos 2
2
cos 2 = cos sin
®“°
cos 2x
sin 2x
2
2
= cos xsin x
= 2 sin x cos x
15 ˆ Ê Ê 1ˆ Á ˜ Ë ¯ = ............................................. Ë 4 4 ¯
Ê 1ˆ Ê 15 ˆ 2 Á ˜ Á = ............................................. Ë 4¯ Ë 4 ˜¯
15 1 = ............................................. 16 16
15 = ............................................. 8
2
2
14 = ............................................. 16 7 = ............................................. 8
4.
®ß· ¥ß«à“ 1) sin (90 A) = cos A
2) cos (90 A) = sin A
«‘∏’∑”
«‘∏’∑”
sin (90 A) = sin 90 cos Acos 90 sin A
cos (90 A)
1cos A0sin A = .....................................................
= cos 90 cos Asin 90 sin A 0cos A1sin A = .....................................................
cos A = .....................................................
sin A = .....................................................
3) tan (90 A) = cot A
4) cot (90 A) = tan A
«‘∏’∑”
«‘∏’∑”
tan (90 A)
o sin (90 A ) o cos (90 A ) cos A = ..................................................... sin A
cot (90 A)
o cos (90 A ) sin (90o A ) sin A = ..................................................... cos A
=
=
cot A = .....................................................
tan A = .....................................................
163
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
5.
5) sec (90 A) = csc A
6) csc (90 A) = sec A
«‘∏’∑”
«‘∏’∑”
sec (90 A)
csc (90 A)
=
1 o cos (90 A ) 1 = ..................................................... sin A
=
1 o sin (90 A ) 1 = ..................................................... cos A
csc A = .....................................................
sec A = .....................................................
7) sin (270 A) = cos A
8) cos (270 A) = sin A
«‘∏’∑”
«‘∏’∑”
sin (270 A)
cos (270 A)
= sin 270 cos Acos 270 sin A
= cos 270 cos Asin 270 sin A
cos A0sin A = (1) .....................................................
0cos A(1)sin A = .....................................................
A = cos .....................................................
sin A = .....................................................
®ßÀ“§à“¢Õß 1) sin 10 cos 50 cos 10 sin 50 = sin (10 50 ) sin 60 = ..................................................... 3 = ..................................................... 2 3) cos 70 cos 50 sin 70 sin 50
2) sin 40 cos 10 cos 40 sin 10 sin (40 10 ) = ..................................................... sin 30 = ..................................................... 1 = ..................................................... 2 4) cos 75 cos 15 sin 75 sin 15
= cos (70 50 )
cos (75 15 ) = .....................................................
cos 120 = ..................................................... 1 = ..................................................... 2
cos 60 = ..................................................... 1 = ..................................................... 2
5) sin
7p 5p 7p 5p cos cos sin 12 12 12 12
6) sin
7p 5p 7p 5p cos cos sin 12 12 12 12
Ê 7p 5pˆ sinÁ ˜ = ..................................................... Ë 12 12 ¯
Ê 7p 5pˆ sin Á ˜ = ..................................................... Ë 12 12 ¯
2p sin = ..................................................... 12
12 p sin = ..................................................... 12
p sin = ..................................................... 6 1 = ..................................................... 2
sin p = ..................................................... 0 = .....................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
164
7) cos
11p 7p 11p 7p cos sin sin 12 12 12 12
8) cos
11p 7p 11p 7p cos sin sin 12 12 12 12
Ê 11p 7 pˆ cos Á ˜ = ..................................................... Ë 12 12 ¯
Ê 11p 7 pˆ cos Á ˜ = ..................................................... Ë 12 12 ¯
18 p cos = ..................................................... 12 3p cos = ..................................................... 2
4p cos = ..................................................... 12 p cos = ..................................................... 3 1 = ..................................................... 2 5p p 5p p 10) cos cos sin sin 6 2 6 2
0 = ..................................................... 9) cos
2p p 2p p cos sin sin 3 6 3 6
Ê 2 p pˆ cos Á ˜ = ..................................................... Ë 3 6¯
Ê 5p pˆ cos Á ˜ = ..................................................... Ë 6 2¯
5p cos = ..................................................... 6
8p cos = ..................................................... 6
pˆ Ê cos Á p ˜ = ..................................................... Ë 6¯
pˆ Ê cos Á p ˜ = ..................................................... Ë 3¯
p cos = ..................................................... 6 3 = ..................................................... 2 p p Ê pˆ Ê pˆ 11) cos cos Á ˜ sin sin Á ˜ Ë 6¯ Ë 6¯ 3 3
p cos = ..................................................... 3 1 = ..................................................... 2 p p p p 12) cos cos sin sin 4 4 4 4
È p Ê pˆ ˘ cos Í Á ˜ ˙ = ..................................................... Î 3 Ë 6¯ ˚
Ê p pˆ cos Á ˜ = ..................................................... Ë 4 4¯
Ê p pˆ cos Á ˜ = ..................................................... Ë 3 6¯
cos 0 = .....................................................
p = cos ..................................................... 2
1 = .....................................................
= 0..................................................... 13) sin
p p p p cos cos sin 12 12 3 3
Êp p ˆ sinÁ ˜ = ..................................................... Ë 3 12¯
3p sin = ..................................................... 12 p sin = ..................................................... 4 2 = ..................................................... 2
14) sin 132 cos 12 cos 132 sin 12 sin (132 12 ) = ..................................................... sin 120 = ..................................................... sin 60 = ..................................................... 3 = ..................................................... 2
165
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
6.
7p 7p ®ßÀ“§à“¢Õß sin 12 ·≈– cos 12 7p Ê p pˆ = sin Á ˜ Ë 3 4¯ 12 p p p p = sin cos cos sin .......................................................................... 3 4 3 4
7p Ê p pˆ = cos Á ˜ Ë 3 4¯ 12 p p p p = cos cos sin sin ..................................................................... 3 4 3 4
3 2 1 2 = .......................................................................... 2 2 2 2
3 2 1 2 = ..................................................................... 2 2 2 2
3 2 2 .......................................................................... = 4
2 3 2 ..................................................................... = 4
2 .......................................................................... = ( 3 1) 4
2 ..................................................................... = (1 3 ) 4
«‘∏’∑”
7.
sin
®ßÀ“§à“¢Õß 1) cos 165
2) sin 165
= cos(120 45 )
sin(120 45 ) = ........................................................
= cos 120 cos 45 sin 120 sin 45
sin 120 cos 45 cos 120 cos 45 = ........................................................
Ê 1 ˆ Ê 2ˆ Ê 3ˆ Ê 2ˆ = ........................................................ Á ˜ Á ˜ Á ˜ Á ˜ Ë 2¯ Ë 2 ¯ Ë 2 ¯ Ë 2 ¯
Ê 3ˆ Ê 2ˆ Ê 1 ˆ Ê 2ˆ = ........................................................ Á ˜ Á ˜ ÁË ˜¯ Á ˜ 2 Ë 2¯ Ë 2 ¯Ë 2 ¯
3 2 2 = ........................................................ 4 4
3 2 2 = ........................................................ 4 4
2 (1 3 ) = ........................................................ 4 3) cos 22.5
2 ( 3 1) = ........................................................ 4 4) sin 22.5
=
8.
cos
o 1 cos 45 2
=
o 1 cos 45 2
2 1 2 = ........................................................ 2
2 1 2 = ........................................................ 2
2 2 = ........................................................ 4
2 2 = ........................................................ 4
1 2 2 = ........................................................ 2
1 2 2 = ........................................................ 2
°”Àπ¥ tan 36 = «‘∏’∑”
10 2 5 5 1
2
tan 36 Ê 10 2 = Á 5 1 Ë
®ßÀ“ cos 72 cos 72
5ˆ ˜ ¯
2
=
2
o
2
o
1 tan 36 1 tan 36
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
166
9.
=
10 2 5 5 2 5 1
=
1 (5 2 5 ) 1 (5 2 5 )
=
10 2 5 6 2 5 62 5 62 5
=
4 2 5 6 2 5 6 2 5 6 2 5
=
60 20 5 12 5 45 36 20
=
24 8 5 12 5 20 36 20
=
80 32 5 = 5 2 5 16
=
4 5 4 16
=
5 ............................................................ 4
............................................................ ............................................................ ............................................................ 1
®ßÀ“§à“¢Õß 1) sin 105
2) sin 135
= sin (60 45 ) = sin 60 cos 45 cos 60 sin 45
= sin (90 45 ) sin 90 cos 45 cos 90 sin 45 = ........................................................
3 2 1 2 = ........................................................ 2 2 2 2
2 2 1 0 = ........................................................ 2 2
2 ( 3 1) = ........................................................ 4
2 = ........................................................ 2
3) sec 225 =
4) cosec 315
1
=
o cos 225
1
o sin 315
= cos 225
= sin 315
= cos (180 45 )
= sin (360 45 ) sin 360 cos 45 cos 360 sin 45 = ........................................................
= cos 180 cos 45 sin 180 sin 45 Ê 2ˆ 2 (1)Á ˜ 0 = ........................................................ 2 Ë 2¯
2 2 1 0 = ........................................................ 2 2 2 = ........................................................ 2 2 2 cos 315 = ............... = ............... 2
2 = ........................................................ 2 2 2 sec 225 = ............... = ............... 2 5) tan 75
6)
Ê 5p ˆ cot Á ˜ Ë 12 ¯
5p 12
= tan (45 30 ) o o tan 45 tan 30 = ........................................................ o o 1 tan 45 tan 30
= cot
1 1 3 = ........................................................ 1 1 3
p p cot cot 1 4 6 = p p cot cot 4 6
Ê p pˆ = cot Á ˜ Ë 4 6¯
167
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 31 31 = ........................................................ 31 31
(1)( 3) 1 = ........................................................ 31
3 2 31 = ........................................................ 31
31 31 = ........................................................ 1 3 31
42 3 = ........................................................ 2
3 2 31 = ........................................................ 31
3 = 2 ........................................................
42 3 = ........................................................ 2 2 3 = ........................................................
7)
5p p tan 3 3 8) p 5p 1 tan tan 3 3 Ê 5p pˆ tanÁ ˜ = ........................................................ Ë 3 3¯
o o tan 10 tan 20 o o 1 tan 10 tan 20
tan
= tan (10 20 ) tan 30 = ........................................................ 3 = ........................................................ 3
4p tan = ........................................................ 3 p tan = ........................................................ 3
3 = ........................................................ 10.
®ßÀ“§à“¢Õß sin «‘∏’∑” ∂â“
®–‰¥â
5p 8
·≈– cos
5p x = 4 5p x = 4
cos
5p 4
5p 8
=
x 5p = 2 8 p p 4
=
3 p p cos(p ) = cos = 4 4 2
·≈â«
2 Ê 5p ˆ sin Á ˜ Ë 8¯
=
㪉 ٵà 2
x 1 = (1cos x) 2 2
2
x 1 = (1cos x) 2 2
cos
sin
1Ê 5p ˆ Á1 cos ˜ 2Ë 4¯
Ê 2ˆ ˘ 1È Í1 Á ˜ ˙ = .......................................................................................... 2 ÍÎ Ë 2 ¯ ˙˚ 1Ê 2ˆ Á1 ˜ = .......................................................................................... 2¯ 2Ë 5p 8
1Ê 2ˆ = 1 2 2 Á1 ˜ = .......................................................................................... 2 2Ë 2¯
2 Ê 5p ˆ cos Á ˜ Ë 8¯
1Ê 5pˆ Á 1 cos ˜ = .......................................................................................... 2Ë 4¯
sin
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
168
Ê 2ˆ ˘ 1È = .......................................................................................... Í1 Á ˜ ˙ 2 ÍÎ Ë 2 ¯ ˙˚
1Ê 2ˆ = .......................................................................................... Á 1 ˜ 2Ë 2¯ cos
11.
°”Àπ¥ sin x =
4 5
5p 8
1 ˆ Ê 1 Á1 2˜ = 2 2 = .......................................................................................... 2 2Ë 2¯
·≈– 0 x
p 2
®ßÀ“
1) sin 2x = 2 sin x cos x
®“° sin2 xcos2 x
2
cos x = 1sin x
Ê 4 ˆ Ê 3ˆ 2Á ˜ Á ˜ = ........................................................ Ë 5¯ Ë 5¯ 24 = ........................................................ 25
= 1
2
cos x =
‡¡◊ËÕ 0 x 2, cos x
2) cos 2x
2
1 sin x
=
1 sin x
=
Ê 4ˆ 1 Á ˜ Ë 5¯
2
2
2 cos x1 = ........................................................
=
2
Ê 3ˆ = ........................................................ 2Á ˜ 1 Ë 5¯
·≈– tan x
18 = ........................................................ 1 25 7 = ........................................................ 25
3) tan 2x =
=
2
3 5 4 3
4) sin 4x
2 tan x 2
1 tan x Ê 4ˆ 2Á ˜ Ë 3¯ = ........................................................ 2 Ê 4ˆ 1 Á ˜ Ë 3¯
8 9 = ........................................................ 3 (7) 24 = ........................................................ 7
= 2 sin 2x cos 2x Ê 24ˆ Ê 7 ˆ 2 Á ˜ Á ˜ = ........................................................ Ë 25¯ Ë 25¯ 336 = ........................................................ 625
169
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
5) cos 4x
6) tan 4x 2
= 2 cos 2x1
=
7 2 2 Ê ˆ 1 Ë 25¯ = ........................................................
98 625 = ........................................................ 625 527 = ........................................................ 625
o 1 cos 180 2 o 180 sin = ........................................................ 2
2
1 tan 2 x
Ê 24ˆ 2Á ˜ Ë 7¯ = ........................................................ 2 Ê 24ˆ 1 Á ˜ Ë 7¯
49 ˆ 2Ê 1 = ........................................................ Ë 625¯
7)
2 tan 2 x
48 49 = ........................................................ 7 (527)
336 = ........................................................ 527 8)
o 1 cos 300 2 o 300 cos = ........................................................ 2
sin 90 = ........................................................
cos 150 = ........................................................
1 = ........................................................
3 = ........................................................ 2 o sin 450 10) o 1 cos 450 o 450 tan = ........................................................ 2 225 = tan ........................................................
o 9) 1 cos 300o 1 cos 300 o 300 = tan 2 150 = tan ........................................................ (180 30 ) = tan ........................................................
(180 45 ) = tan ........................................................
30 ) = (tan ........................................................ 1 = ........................................................ 3 o 1 2 tan 22 2 11) o 2 1 1 tan 22 2
45 = tan ........................................................ = 1........................................................
12)
sin 420
o
1 cos 420
o
o 1 = tan 2 (22 ) 2 tan 45 = ........................................................
o 420 tan = ........................................................ 2
1 = ........................................................
tan (180 30 ) = ........................................................
tan 210 = ........................................................ tan 30 = ........................................................ 1 = ........................................................ 3
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
170
12.
4 5
∂â“ 90 x 180 ·≈– sin x =
®ßÀ“
1) cos x
2) tan x cos x =
«‘∏’∑”
2
1 sin x
tan x =
«‘∏’∑”
4 5 = ............................... 3 5
2
1 ÊÁ 4ˆ˜ = ............................... Ë 5¯
3 = ............................... 5 3) cos
x 2
4) sin
3 5 3 1 5
x 2 x cos 2 2
cos
5) tan
«‘∏’∑”
x 1 2 2 x = 2 cos 1 2 2 x = 2 cos 2
cos x = 2 cos
«‘∏’∑”
2
x 2
x x cos 2 2 x 1 2 sin = ............................... 2 5
sin x = 2 sin
«‘∏’∑”
4 5 x 4 5 sin .................. = ............................... 2 5 2
1 5 1 = ............................... 5
2 5 = ............................... 5
6) cos
x tan 4
4 = ............................... 3
=
x 4 =
1 cos
x 2
x sin 2
1 1 5 = ............................... 2 5 5 51 5 = ............................... 5 2 5
51 = ............................... 2
sin x cos x
«‘∏’∑”
x 4
cos
2 cos
2
x 2
= 2 cos
x 4
= cos
2
x 1 4
x 1 2
1 1 = ............................... 5 51 = ............................... 5 2 x 51 5 cos .................. = ............................... 4 2 5 5 2 x 5 5 cos .................. = ............................... 4 10
x 5 5 cos .................. = ............................... 4 10 1 = ............................... 50 10 5 10
171
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 13.
®ßÀ“§à“¢Õß p p cos 8 8
1) 2 sin
2) cos
Ê pˆ = 2 sin 2 Á ˜ Ë 8¯ = sin
2
2 p p sin 12 12
Ê pˆ = cos 2 Á ˜ Ë 12¯
p 4
= cos
2 = ........................................................ 2
p 6
3 = ........................................................ 2
2
2
2 cos 15 1
12 sin 75
cos 2(15 ) = ........................................................
cos 2 (75 ) = ........................................................
cos 30 = ........................................................
cos 150 = cos 30 = ........................................................
3 = ........................................................ 2 o 1 2 tan 22 2 3) o 2 1 1 tan 22 2
3 = ........................................................ 2
o Ê 22 1 ˆ tan 2 = ........................................................ Ë 2¯ tan 45 = ........................................................ 1 = ........................................................
4)
o tan 75 o 2 1 1 tan 75 2 2 =
o tan 75
o 2 1 (1 tan 75 ) 2 o 2 tan 75 = 2 o 1 tan 75
tan 2 (75 ) = ........................................................ tan 150 = tan (180 30) = ........................................................
5) sin 75 sin 15
3 = ........................................................ 3 6) sin 75 sin 15
Ê 75o 15o ˆ Ê 75o 15o ˆ = 2 sin Á ˜ cos Á ˜ 2 2 Ë ¯ Ë ¯
Ê 75o 15o ˆ Ê 75o 15o ˆ = 2 cos Á ˜ sin Á ˜ 2 2 Ë ¯ Ë ¯
= 2 sin 45 cos 30
2 cos 45 sin 30 = ........................................................
2 3 2 = ........................................................ 2 2
2 1 2 = ........................................................ 2 2
6 = ........................................................ 2 sin 45 sin 15 Ê 45o 15o ˆ Ê 45o 15o ˆ 2 sin cos = ........................................................ Á ˜ Á ˜ 2 2 Ë ¯ Ë ¯
2 = ........................................................ 2 sin 45 sin 15 Ê 45o 15o ˆ Ê 45o 15o ˆ 2 cos sin = ........................................................ Á ˜ Á ˜ 2 2 Ë ¯ Ë ¯
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
172 2 sin 30 cos 15 = ........................................................
2 cos 30 sin 15 = ........................................................
1 31 2 = ........................................................ 2 2 2
3 31 2 = ........................................................ 2 2 2
2 ( 3 1) = ........................................................ 4
6 ( 3 1) = ........................................................ 4 8) cos 75 cos 15
7) cos 75 cos 15
Ê 75o 15o ˆ Ê 75o 15o ˆ = 2 cos Á ˜ cos Á ˜ 2 2 Ë ¯ Ë ¯
Ê 75o 15o ˆ Ê 75o 15o ˆ = 2 sin Á ˜ sin Á ˜ 2 2 Ë ¯ Ë ¯
2 cos 45 cos 30 = ........................................................
2 sin 45 sin 30 = ........................................................
2 1 2 = ........................................................ 2 2
2 1 2 = ........................................................ 2 2
2 = ........................................................ 2
2 = ........................................................ 2
9) cos 45 cos 15
14.
10) cos 45 cos 15
Ê 45o 15o ˆ Ê 45o 15o ˆ 2 cos Á ˜ cos Á ˜ = ........................................................ 2 2 Ë ¯ Ë ¯
Ê 45o 15o ˆ Ê 45o 15o ˆ 2 sin Á s in = ........................................................ ˜ Á ˜ 2 2 Ë ¯ Ë ¯
cos 30 cos 15 = 2........................................................
2 sin 30 sin 15 = ........................................................
1 31 2 = ........................................................ 2 2 2
1 31 2 = ........................................................ 2 2 2
31 = ........................................................ 2 2
2 ( 3 1) = ........................................................ 4
®ßÀ“§à“¢Õß 1) sin 45 cos 15
2) cos 45 sin 15
=
1 [sin (45 15 )sin (45 15 )] 2 1 = (sin 60 sin 30 ) 2
=
1 Ê 3 1ˆ ˜ = ....................................................... Á 2Ë 2 2¯
1 Ê 3 1ˆ ˜ = ........................................................ Á 2Ë 2 2¯
3 1 = ....................................................... 4
31 = ........................................................ 4
3) sin 75 cos 15
4) cos 75 sin 15
1 sin (75 15 )sin (75 15 ) = ........................................................ 2 1 (sin 90 sin 60 ) = ........................................................ 2
[
1 [sin (45 15 )sin (45 15 )] 2 1 = ........................................................ (sin 60 sin 30 ) 2
]
1 sin (75 15 )sin (75 15 ) = ........................................................ 2 1 (sin 90 sin 60 ) = ........................................................ 2
[
]
173
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 3ˆ 1Ê = ........................................................ Á1 ˜ 2Ë 2¯
3ˆ 1Ê = ........................................................ Á1 ˜ 2Ë 2¯
2 3 = ........................................................ 4
2 3 = ........................................................ 4
5) cos 45 cos 15
6) sin 45 sin 15
=
1 [cos (45 15 )cos (45 15 )] 2 1 (cos 60 cos 30 ) = ..................................................... 2
1 [cos (45 15 )cos (45 15 )] 2 1 (cos 30 cos 60 ) = ........................................................ 2
1Ê1 3ˆ = ..................................................... 2Ë2 2 ¯
1 Ê 3 1ˆ = ........................................................ ˜ Á 2Ë 2 2¯
1 3 = ..................................................... 4
31 = ........................................................ 4
7) cos 75 cos 15 1 cos (75 15 )cos (75 15 ) = ........................................................ 2 1 (cos 90 cos 60 ) 2 = ........................................................
[
15.
=
8) sin 75 sin 15
]
=
1 [cos (75 15 )cos (75 15 )] 2
1 (cos 60 cos 90 ) = ........................................................ 2
1Ê 1ˆ Á0 ˜ 2Ë 2¯ = ........................................................
1Ê 1 ˆ Á 0˜ = ........................................................ ¯ 2Ë 2
1 4 = ........................................................
1 = ........................................................ 4
°”Àπ¥
4 p 0 5 2 5 p cos = 0 13 2
sin =
,
,
Y B(5, 12)
12
Y A(3, 4)
4
O
3
X
O
5
X
®ßÀ“ 1) sin ( )
2) sin ( )
= sin cos cos sin
sin cos cos sin = ........................................................
Ê 4ˆ Ê 5 ˆ Ê 3ˆ Ê 12ˆ = Á ˜Á ˜ Á ˜Á ˜ Ë 5¯ Ë 13¯ Ë 5¯ Ë 13¯
Ê 4ˆ Ê 5 ˆ Ê 3ˆ Ê 12ˆ Á ˜Á ˜ Á ˜Á ˜ = ........................................................ Ë 5¯ Ë 13¯ Ë 5¯ Ë 13¯
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
174
=
20 36 65 65
56 = ........................................................ 65
3) cos ( )
16.
20 36 = ........................................................ 65 65 16 = ........................................................ 65 4) cos ( )
= cos cos sin sin
cos cos sin sin = ........................................................
Ê 3ˆ Ê 5 ˆ Ê 4ˆ Ê 12ˆ = Á ˜Á ˜ Á ˜Á ˜ Ë 5¯ Ë 13¯ Ë 5¯ Ë 13¯ 15 48 = ........................................................ 65 65 33 = ........................................................ 65
Ê 3ˆ Ê 5 ˆ Ê 4ˆ Ê 12ˆ = ........................................................ Ë 5¯ Ë 13¯ ÁË 5˜¯ ÁË 13˜¯
°”Àπ¥
p p 2
cos =
5 3
sin =
7 3p , 2 2p 4
,
15 48 = ........................................................ 65 65 63 = ........................................................ 65
Y
Y
( 5 , 2) 3 2 5
3
O
X
O
X
4 (3, 7 )
®ßÀ“ 1) sin ( )
2) sin ( )
sin cos cos sin = ........................................................
= sin cos cos sin
Ê 2ˆ Ê 3ˆ Ê 5 ˆ Ê 7ˆ = ........................................................ Ë 3¯ Ë 4¯ ÁË 3 ˜¯ ÁË 4 ˜¯
Ê 2ˆ Ê 3ˆ Ê 5 ˆ Ê 7ˆ = Á ˜ Á ˜ Á ˜ Á ˜ Ë 3¯ Ë 4 ¯ Ë 3 ¯ Ë 4 ¯
35 6 = ........................................................ 12 12
35 6 = ........................................................ 12 12
6 35 = ........................................................ 12 3) cos ( )
6 35 = ........................................................ 12 4) cos ( )
cos cos sin sin = ........................................................
= cos cos sin sin
Ê 5 ˆ Ê 3ˆ Ê 2 ˆ Ê 7ˆ Á ˜ ÁË ˜¯ ÁË ˜¯ Á ˜ = ........................................................ 3 Ë 4¯ Ë 3¯ 4
Ê 5 ˆ Ê 3ˆ Ê 2 ˆ Ê 7ˆ = ........................................................ Á ˜ ÁË ˜¯ ÁË ˜¯ Á ˜ 3 Ë 4¯ Ë 3¯ 4
3 5 2 7 = ........................................................ 12 12
3 5 2 7 = ........................................................ 12 12
3 5 2 7 = ........................................................ 12
3 5 2 7 = ........................................................ 12
175
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
17.
∂â“ sin x = 35 , sin (xy) = 135 ‚¥¬∑’Ë 0 x ®ßÀ“ 1) sin y «‘∏’∑” ‡π◊ËÕß®“° sin x =
®“°
·≈– p xy
3p 2
2) tan (xy) 3 5
·≈– 0 x cos x =
¥—ßπ—Èπ
p 2
p 2
Ê 3ˆ 1 Á ˜ Ë 5¯
2
4 = ................. 5
5 13 5 sin x cos ycos x sin y = 13 3 4 5 cos y sin y = 5 5 13 3p p xy 2
sin (xy) =
.....(1)
‡π◊ËÕß®“° ¥—ßπ—Èπ
cos (xy) = 1 sin ( x y) 2
2
1 ÊÁ 5 ˆ˜ = ........................................................................... Ë 13¯ 12 = ........................................................................... 13 12 cos (xy) = 13 12 cos x cos ysin x sin y = 13 4 3 12 cos y sin y = ......................................................................................................................... .....(2) 5 5 13 12 16 20 cos y sin y = (1)4; ......................................................................................................................... .....(3) 5 5 13 12 9 36 cos x sin y = (2)3; ......................................................................................................................... .....(4) 5 5 13 16 5 sin y = (3)(4) ......................................................................................................................... 13 16 sin y = ......................................................................................................................... 65 sin (x y) tan (x+y) = cos ( x y)
®“°
5 13 = ........................................................................... 12 13
5 = ........................................................................... 12
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
176
18.
∂â“
p 3p x p p y 2 2
,
·≈– cos x = 35 , tan y =
5 12
®ßÀ“ cos (xy) «‘∏’∑” ®“° p2 x p ·≈– cos x = 35 sin x =
¥—ßπ—Èπ
2
1 cos x 2
Ê 3ˆ 1 Á ˜ = ........................................................................... Ë 5¯
·≈– p y
4 = ........................................................................... 5
3p 2
¥—ßπ—Èπ sin y 0 ·≈– cos y 0 ®“° tan y = sin y cos y
®–‰¥â
=
5 12 5 13 12 13
cos x cos ysin x sin y cos (xy) = ........................................................................... Ê 3ˆ Ê 12ˆ Ê 4ˆ Ê 5 ˆ = ........................................................................... Á ˜ Á ˜ Á ˜ Á ˜ Ë 5¯ Ë 13¯ Ë 5¯ Ë 13¯ 36 20 = ........................................................................... 65 65 56 = ........................................................................... 65
19.
®ßÀ“§à“¢Õß 2
1) (sin xcos x) sin 2x 2
2
= sin x2 sin x cos xcos x2 sin x cos x 2
2
= (sin xcos x)(2 sin x cos x2 sin x cos x) = 10 = 2)
............................................................ 1 ............................................................
2 sin 2 x sin x cos x 2 cos x
4
4
2
3) sec xtan x2 tan x
2
2 2 sin x cos x cos x 2 cos x
= (sec xtan x)(sec xtan x)2 tan x
= sin xcos x
= 1(sec xtan x)2 tan x
=
2 2 ............................................................ 1 ............................................................
=
=
2
2
2
=
2
2
2
2
............................................................ 2 2 sec xtan x ............................................................ 1 ............................................................
2
177
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
4)
Èsin 2 x cos 2 x ˘ sec x tan2 x cos x ˚˙ ÎÍ sin x
5) cos 2xsin 2x tan x
2 sin 2 x cos x cos 2 x sin x ˘ sin x = È sec x tan x = cos 2xsin 2x ÍÎ ˙ cos x sin x cos x ˚
È sin (2x x )˘ sec x 2 = ............................................................ tan x ÍÎ sin x cos x ˙˚
cos 2 x cos x sin 2 x sin x = ............................................................ cos x cos (2x x ) = ............................................................ cos x cos x = ............................................................ cos x = ............................................................ 1
È sin x ˘ sec x 2 = ............................................................ tan x ÍÎ sin x cos x ˙˚ 2
= ............................................................ sec xsec xtan x 2
2
= ............................................................ sec xtan x = ............................................................ 1
2.8 µ—«º°º—π¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘ øíß°å™—π∑’Ë¡’µ—«º°º—π‡ªìπøíß°å™—π øíß°å™—ππ—ÈπµâÕ߇ªìπøíß°å™—π 1-1 ‡∑à“π—Èπ ‡π◊ËÕß®“° øíß°å™—πµ√’ ‚°≥¡‘µ‘‰¡à‡ªìπøíß°å™—π 1-1 ¥—ßπ—Èπµ—«º°º—π¢Õßøíß°å™—πµ√’ ‚°≥¡‘µ‘®÷߉¡à‡ªìπøíß°å™—π ·µà ∂â“°”À𥂥‡¡π¢Õßøíß°å™π— µ√’ ‚°≥¡‘µ„‘ Àâ‡À¡“– ¡ µ—«º°º—π¢Õßøíß°å™π— µ√’ ‚°≥¡‘µ®‘ –‡ªìπøíß°å™π—
2.8.1 µ—«º°º—π¢Õßøíß°å™—π‰´πå øíß°å™—π {(x, y)|y = sin x }, x , 1 y 1 ‰¡à‡ªìπøíß°å™—π 1-1 ¥—ßπ—Èπ µ—«º°º—π¢Õßøíß°å™—π‰´πå§◊Õ {(x, y)|x = sin y} ®÷߉¡à‡ªìπøíß°å™—π ·µà∂â“°”À𥂥‡¡π¢Õßøíß°å™—π‰´π凪ìπ x| p2 x p2 øíß°å™—π (x, y)|y = sin x,
{
p p x 2 2
‡√’¬°«à“
}
} ‡ªìπøíß°å™—π - ®–¡’øíß°å™—πº°º—π§◊Õ { , 1 1
{
(x y) x = sin y
|
,
p p y 2 2
} ´÷Ëß
arcsine
∫∑𑬓¡ øíß°å™—π arcsine §◊Õ‡´µ¢ÕߧŸàÕ—π¥—∫ (x, y) ‚¥¬∑’Ë x = sin y ·≈– p2
y
p 2
‡¡◊ËÕ (x, y) arcsine ®–‰¥â y = arcsine x À√◊Õ y = arcsin x ¡’§«“¡À¡“¬‡™àπ‡¥’¬«°—∫ p p x = sin y ‡¡◊ËÕ y 2 2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
178
y = arcsin x 1 x 1
,
·≈– p2
p 2
y
‚¥‡¡π¢Õßøíß°å™—π arcsine §◊Õ [1, 1] ·≈–‡√π®å¢Õßøíß°å™—π arcsine §◊Õ Y
Y
π 2
Ê1, p ˆ Ë 2¯y = x Ê p , 1ˆ Ë2 ¯
π 2
Ê p , 1ˆ Ë2 ¯
1
1 y = sin x
π 1 2
Ê p , 1ˆ Ë 2 ¯
È p , p ˘ ÎÍ 2 2 ˙˚
O
1
y = sin x
X
π 2
1 π 2
π 1 2 Ê p , 1ˆ Ë 2 ¯
1 π Ê1, p ˆ 2 Ë 2¯
π 2 y = arcsin x
O
1
X
µ—«Õ¬à“ß∑’Ë 1 ®ßÀ“§à“¢Õß
«‘∏’∑”
1 2
1)
arcsin
1)
„Àâ arcsin ‡π◊ËÕß®“°
2)
2) arcsin (1)
1 = 2 p p 2 2
„Àâ arcsin (1) = ‡π◊ËÕß®“° p2
p 2
1 2
®–‰¥â
sin =
®–‰¥â
sin = sin
¥—ßπ—Èπ
=
π—Ëπ§◊Õ arcsin 12 ®–‰¥â sin ®–‰¥â sin
=
p 6 p 6
π—Ëπ§◊Õ
µÕ∫
= 1 = 1
sin = sin
¥—ßπ—Èπ
p (¡’‡æ’¬ß§à“‡¥’¬«) 6
3p 2 3p arcsin (1) = 2
3p (¡’‡æ’¬ß§à“‡¥’¬«) 2
=
µÕ∫
179
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
2.8.2 µ—«º°º—π¢Õßøíß°å™—π‚§‰´πå øíß°å™—π {(x, y)|y = cos x}, x , 1 y 1 ‰¡à‡ªìπøíß°å™—π 1-1 ¥—ßπ—Èπ µ—«º°º—π¢Õßøíß°å™—π‚§‰´πå§◊Õ {(x, y)|x = cos y} ®÷߉¡à‡ªìπøíß°å™—π ‡¡◊ËÕ°”À𥂥‡¡π¢Õß y = cos x ‚¥¬„Àâ 0 x p ®–‰¥âøíß°å™—π {(x, y)|y = cos, 0 x p} ‡ªìπøíß°å™—π 1-1 ´÷Ëß¡’øíß°å™—πº°º—π {(x, y)|x = cos y, 1 x 1, 0 y p} ‡√’¬° øíß°å™—πº°º—ππ’È«à“ arccosine ∫∑𑬓¡ øíß°å™—π arccosine §◊Õ‡´µ¢ÕߧŸàÕ—π¥—∫ (x, y) ‚¥¬∑’Ë x = cos y ·≈– 0 y p ‡¡◊ËÕ (x, y) arccosine ®–‰¥â y = arccosine x À√◊Õ y = arccos x ¡’§«“¡À¡“¬‡™àπ‡¥’¬«°—∫ x = cos y ‡¡◊ËÕ 0 y p y = arccos x, 1 x 1 ·≈– 0 y p ‚¥‡¡π¢Õßøíß°å™—π arccos §◊Õ [1, 1] ·≈–‡√π®å¢Õßøíß°å™—π arccos §◊Õ [0, p] (1, π)
Y (0, 1)
y arccos x
y = cos x
O
π
yx
π 2 (0, 1)
π 2
1
Y
X
π
(π, 1)
O
1
π (1, 0) 2
1
y cos x
π
X (π, 1)
µ—«Õ¬à“ß∑’Ë 2 ®ßÀ“§à“¢Õß «‘∏’∑”
1)
arccos 1
1)
„Àâ arccos 1 = ‡π◊ËÕß®“° 0 p
1 2) arccos Ê ˆ Ë 2¯
®–‰¥â cos = 1 ®–‰¥â cos = cos 0 ¥—ßπ—Èπ = 0 π—Ëπ§◊Õ arccos 1 = 0
µÕ∫
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
180
2)
Ê 1 ˆ = Ë 2¯
®–‰¥â
cos =
‡π◊ËÕß®“° 0 p
®–‰¥â
cos = cos
„Àâ arccos
π—Ëπ§◊Õ arccos
= Ê 1 ˆ Ë 2¯
2p 3
3p ( 2 2p 3
sin =
¥—ßπ—Èπ
1 2
¡’æ’¬ß§à“‡¥’¬«)
2p 3
=
µÕ∫
2.8.3 µ—«º°º—π¢Õßøíß°å™—π·∑π‡®πµå ∫∑𑬓¡ øíß°å™—π arctangent §◊Õ‡´µ¢ÕߧŸàÕ—π¥—∫ (x, y) ‚¥¬∑’Ë x = tan y ·≈– p2 ‡¡◊ËÕ (x,
y) arctangent
‡¥’¬«°—∫ x = tan y ·≈– p2
®–‰¥â
y
y = arctangent x
À√◊Õ
y = arctan x
Y π 2
1 O
X
π 2
1
O
1
X
1
π y arctan x 2
y tan x y tan x
p p x 2 2
·≈– y
x ·≈–
p p y 2 2
‚¥‡¡π¢Õßøíß°å™—π arctan §◊Õ R ·≈–‡√π®å¢Õßøíß°å™—π arctan §◊Õ µ—«Õ¬à“ß∑’Ë 3 ®ßÀ“§à“¢Õß «‘∏’∑”
1)
arctan
1)
„Àâ arctan 3 = ‡π◊ËÕß®“° p2
2) arctan (1)
3
p 2
®–‰¥â ®–‰¥â
tan =
3
tan = tan
p 3
p 2
¡’§«“¡À¡“¬‡™àπ
p 2
Y
π 2
y
Ê p , p ˆ Ë 2 2¯
181
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
π—Ëπ§◊Õ arctan 2)
„Àâ arctan (1) = ‡π◊ËÕß®“° p2
p 2
p 3 p 3 = 3
=
¥—ßπ—Èπ
µÕ∫
®–‰¥â
tan = 1
®–‰¥â
Ê pˆ tan = tan Á ˜ Ë 4¯
¥—ßπ—Èπ
p = 4
π—Ëπ§◊Õ arctan (1) =
p 4
µÕ∫
°‘®°√√¡∑’Ë 2.8 1.
®ßÀ“§à“¢Õß 1) arcsin
3 2 p p 3 = , 2 2 2
„Àâ arcsin ®–‰¥â
2) arcsin 0
3 2
sin =
sin = sin
3) arccos
............................
3 2
„Àâ arccos ®–‰¥â
p 6
p 3 = 4 2
,
cos =
3 2 p cos 6 p 6
cos =
............................
=
............................
¥—ßπ—Èπ arccos
p 3 = 6 2
..........................
p 2
sin =
sin 0 ............................
=
0 ............................
0 ¥—ßπ—Èπ arcsin 0 = ............................ 4) arccos
3 = 0 p 2
sin = 0
®–‰¥â
p ............................ 4
=
¥—ßπ—Èπ arcsin
„Àâ arcsin 0 = , p2
1 2
„Àâ arccos ®–‰¥â
1 = , 0 p 2 1 2 p cos cos = ............................ 3 p = ............................ 3 cos =
¥—ßπ—Èπ arccos
p 1 = 3 2
............................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
182
Ê 2ˆ 6) arcsin Á ˜ Ë 2¯
5) arccos (1)
„Àâ arccos (1) = , 0 p ®–‰¥â
cos =
1 ............................
cos =
cos p ............................
=
p ............................
„Àâ arcsin ®–‰¥â
Ê 2ˆ p p Á ˜ = 2 2 Ë 2¯ 2 sin = 2
,
............................
sin =
p p p ............................ 4 2 2
p ¥—ßπ—Èπ arccos (1) = ........................
=
¥—ßπ—Èπ arcsin 7) arctan
3 3
„Àâ arctan ®–‰¥â
p p 3 = , 3 2 2
3 3
tan =
p 2ˆ Ê = Ë 2¯ 4
..........................
( 3) = ,
3 = 3
p p 2 2
p tan = 3 0 2 Ê pˆ tan = tan Ë 3 ¯ ,
......,
...................................
p p p ................................... 3 2 2
p ........................ 6
=
¥—ßπ—Èπ arctan
„Àâ arctan ®–‰¥â
....................
p ........................ 6
=
¥—ßπ—Èπ arctan
p ................................... 3
p ( 3) = ............................... 3
1 10) arcsin ( ) 2
3 2
„Àâ arccos
p ............................ 4
8) arctan ( 3)
p tan = tan 6
9) arccos
Ê pˆ
sin Á ˜ , ............................ Ë 4¯
3 = 0 p 2 3 cos = 2 p cos = cos 6 p = 6
,
®–‰¥â ....................................................
„Àâ arcsin ( 12 ) = , p2 p2 1 p sin = , 0 ®–‰¥â ........................................................... 2 2
.................................................................
Ê pˆ sin = sin Á ˜ , ......................................................................... Ë 6¯
.................................................................
p p ......................................................................... 0
¥—ßπ—Èπ arccos 23 = p6 .................................................................
.........................................................................
2 6 p = 6 1 p arcsin ( ) = 2 6
¥—ßπ—Èπ .........................................................................
183
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 Ê 2ˆ 12) arccos Á ˜ Ë 2¯
11) arctan 1
„Àâ arctan 1 = , p2
p 2
tan = 1 ®–‰¥â ....................................................
p
tan = tan ................................................................. 4
p
2.
„Àâ arccos
Ê 2ˆ Á ˜ = 0 p Ë 2¯ 2 cos = 2 3p 3p cos = cos ,0 p 4 4
,
®–‰¥â ...........................................................
......................................................................... 3p
= ................................................................. 4
= ......................................................................... 4
¥—................................................................. ßπ—Èπ arctan 1 = p4
¥—......................................................................... ßπ—Èπ arccos ÊÁË 22ˆ˜¯ = 34p
®ßÀ“§à“¢Õß 1) arcsin (cos
p ) 4
2) arccos (sin
Ê 2ˆ = arcsin Á ˜ Ë 2¯ = arcsin (sin =
p ................................................. 4
3) arctan (tan =
p ) 4
p ) 3
p ................................................. 3
p ) 2
= arccos (1) = arccos (cos 0) =
................................................. 0 .................................................
4) arccot (tan
p ) 6
Ê 1ˆ = arccot Á ˜ Ë 3¯ = arccot (cot
p
) ................................................. 3
= 5) arcsec (tan
p ) 4
p ................................................. 3
6) arccosec (cot
3p ) 4 p ) 4
= arcsec (1)
= arccosec (cot
= arcsec (sec 0)
= arccosec (1)
= 0
p = arccosec cosec ( 2 )
................................................. .................................................
[ ] ................................................. p ................................................. 2
=
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
184 3 7) tan Ê arcsin ˆ Ë 2¯
1 8) sec Ê arccos ˆ Ë 2¯
= tan arcsin (sin
[
p = tan 3
=
p ) 3
[
[
[
=
]
p = sin arctan (tan( ) 4 p sin ( ) = 4
p ................................................. 3 2.................................................
[
]
10) cos arccot ( 3 )
= cos arccot(cot
]
[
p = cos 6
.................................................
p )] 6
................................................. 3 .................................................
= 2 2
=
.................................................
3.
]
= sec
................................................. 3 .................................................
9) sin arctan (1)
p ) 3
= sec arccos (cos
]
2
®ßÀ“§à“µàÕ‰ªπ’È ‚¥¬„™âµ“√“ß 1) arctan 6.4348
2) arctan 0.8391
„Àâ arctan 6.4348 = , p2 ®–‰¥â
p 2
tan = tan 1.4166
= 1.4166 .........................
¥—............................................................ ßπ—Èπ arctan 6.4348 = 1.4166 3) arcsin 0.6157
®–‰¥â
p 2
tan = 0.8391 ®–‰¥â .......................................................... tan = tan 0.6981 ...................................................................... = 0.6981 ...................................................................... ¥—............................................................... ßπ—Èπ arctan 0.8391 = 0.6981
tan = 6.4348
„Àâ arcsin 0.6157 = , p2
„Àâ arctan 0.8391 = , p2
4) arcsin 0.8192
0.6157 sin = .........................
p 2
„Àâ arcsin 0.8192 = , p2 ®–‰¥â
p 2
0.8192 sin = ...........................
sin = sin 0.6632 ............................................................
sin = sin 0.9599 ...............................................................
= 0.6632 ............................................................
= 0.9599 ...............................................................
arcsin 0.6157 = 0.6632 ............................................................
¥—............................................................... ßπ—Èπ arcsin 0.8192 = 0.9599
¥—ßπ—Èπ
5) arccos 0.9336
6) arccos 0.7071
„Àâ arccos 0.9336 = , 0 p 0.9336 ®–‰¥â cos = .........................
„Àâ arccos 0.7071 = , 0 p 0.7071 ®–‰¥â cos = ...........................
cos = cos 0.3665 ............................................................
cos = cos 0.7854 ...............................................................
= 0.3665 ............................................................
= 0.7854 ...............................................................
arccos 0.9336 = 0.3665 ............................................................
¥—............................................................... ßπ—Èπ arccos 0.7071 = 0.7854
¥—ßπ—Èπ
185
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 4.
®ßÀ“§à“¢Õß 2 1) sin [arccos ] 3
[
2) csc arccos
2 = 3 2 cos = 0 p 3
«‘∏’∑” „Àâ arccos ®–‰¥â
,
2
3
O
=
5
X
2
1 0 p 3 1 csc arccos 3
Y
[
,
]
2
3
............................... 5 ............................... 9
=
1 = 3
cos =
®–‰¥â
2
Ê 2ˆ 1 Á ˜ Ë 3¯
]
«‘∏’∑” „Àâ arccos
sin = 1cos
Y
1 3
X
1
O
= csc
8
·µà 0 p2 ¥—ßπ—Èπ sin = 35 ............................................................................
=
3 .................... 8
=
3 2 .................... 4
π—Ëπ§◊Õ sin [arccos 23 ] = 35 ............................................................................ [
p ) 6 p arcsin (cos ) = 6 p sin = cos 6
]
3) tan arcsin (cos
«‘∏’∑” „Àâ
sin =
[
5) tan arcsin (cos
p ) 6 p arccos (sin ) = 6
«‘∏’∑” „Àâ
]
cos = sin
3 2
cos =
p .................... 3
cos = cos
p .................... 3
=
p )] 6
[
6) tan arccos (sin
= tan p = tan 3
= tan p = tan 3
=
=
.................... 3 ....................
p .................... 3
p )] 6
.................... 3 ....................
7) tan (arcsin 0.5592)
8) cos (arctan 1.8040)
«‘∏’∑” „Àâ arcsin 0.5592 =
«‘∏’∑” „Àâ arctan 1.8040 =
sin = 0.5592 (
p 6
1 2
p 3
sin = sin
=
[
4) tan arccos (sin
®“°µ“√“ß)
tan = 1.8040
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
186
5.
sin = sin 34
tan = tan 61
= 34
= 61
tan (arcsin 0.5592)
cos (arctan 1.8040)
= tan
= cos
= tan 34
= cos 61
=
=
.................... 0.6745 .................... (®“°µ“√“ß)
®ßÀ“§à“¢Õß 1 1) cos [arcsin ] 2
[
2) sin arccos
1 p p = 2 2 2 1 sin = 2 p sin = sin 6 p = 6 1 cos arcsin ) = cos 2 p = cos 6 3 = 2
,
«‘∏’∑” „Àâ arcsin
.................... 0.4848 .................... 1 2
]
«‘∏’∑” „Àâ arccos
1 = 0 p 2 1 cos = 2 p cos = cos 3 p = 3
,
.................... ....................
[
[
sin arccos
1 2
]
....................
[
3) cos arccos
1 2
]
[
4) sin arcsin
1 = 0 p 2 1 cos = 2 p cos = cos 3 p = 3 1 cos arccos = cos 2 p = cos 3 = 1 2
«‘∏’∑” „Àâ arccos
, .........................
........................ ........................ ........................
[
]
........................ ........................ ........................
1 2
=
cos ....................
=
cos .................... 3
=
1 .................... 2
p
]
p p 1 = 2 2 2 1 sin = 2 p sin = sin 6 p = 6 1 sin arcsin = sin 2 p = sin 6 = 1 2
«‘∏’∑” „Àâ arcsin
, .............................. ........................ ........................ ........................
[
]
........................ ........................ ........................
187
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
5) cos arcsin (
[
3 )] 2
6) sin arccos (
[
«‘∏’∑” „Àâ arcsin ( 23 ) =
«‘∏’∑” „Àâ arccos ( 23 ) =
p p 2 2
®–‰¥â
[
0 p
sin =
3 ..................... 2
sin =
sin( ) ..................... 3
=
..................... 3
cos arcsin (
3 )] 2
p
]
cos =
3 ..................... 2
cos =
cos ..................... 6
=
..................... 6
p
p = cos ( ) 3 1 = 2
3 ) 2
®–‰¥â
..............
sin arccos (
[
3 ) 2
]
.....................
[
7) tan arcsin
1 ] 2
[
8) tan arccos
1 p p = , 2 2 2 1 ®–‰¥â sin = ......................... 2 p sin = ......................... 6 p = ......................... 6 p 1 tan tan [arcsin ] = ......................... 6 2
«‘∏’∑” „Àâ arcsin
=
[ ] «‘∏’∑” „Àâ arctan = 1, p2
[
5p
5p
=
sin ..................... 6
=
1 ..................... 2
1 ] 2
1 = , 0 p 2 1 ®–‰¥â cos = .............................. 2 p cos = .............................. 3 p = .............................. 3 p 1 tan tan [arccos ] = .............................. 3 2
«‘∏’∑” „Àâ arccos
3 .........................
=
3
9) tan arctan 1
5p
[ ] «‘∏’∑” „Àâ arctan 1 = , p2
3 ..............................
10) cot arctan 1
p 2
p 2
tan =
1 ..............................
tan =
1 ..............................
=
p tan .............................. 4
=
tan .............................. 4
=
p .............................. 4
=
.............................. 4
=
p tan .............................. 4
=
cot .............................. 4
=
1 ..............................
=
1 ..............................
]
tan arctan 1
[
]
cot arctan 1
p
p
p
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
188
[
11) sin arccos
3 5
]
«‘∏’∑” „Àâ arccos
[
12) cos arcsin 3 = 5
«‘∏’∑” „Àâ arcsin 3 5
=
®–‰¥â cos
4 5
·≈–
®–‰¥â
] 4 = 5
sin =
0 p 2
2
2
sin = 1cos Ê 3ˆ = 1 Á ˜ Ë 5¯
¥—ßπ—Èπ
sin =
[
«‘∏’∑” „Àâ
®–‰¥â
4
.......... 5
2 ) = 3 2 .......... ·≈– p2 3
sin =
p 2
p 2
0
À“ cot ‚¥¬æ‘®“√≥“®“° 2
=
9 .............................. 25
cos =
3 .............................. 5
π—Ëπ§◊Õ cos [arcsin 45 ] = 53 ......................................................................
]
‡π◊ËÕß®“° sin 0 ·≈– p2 ¥—ßπ—Èπ p2
2
·µà 0 p ...................................................................... ¥—ßπ—Èπ cos = 53 ......................................................................
2 ) 3
arcsin (
p p 2 2 2
4 π—Ëπ§◊Õ sin [arccos 35 ] = .......... 5 13) cot arcsin (
Ê 4ˆ 1 Á ˜ = .............................. Ë 5¯
2
16 25 4 sin = 5
p 2
·≈–
cos = 1sin
=
ᵈ 0
4 5
2
cot = cosec 1 2
Ê 3ˆ Á ˜ 1 Ë 2¯
=
..................................................
=
9 1 .................................................. 2
=
7 .................................................. 2
cot =
7 = .................................................. 2 2
14
189
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
14)
[
]
sin arctan (3)
«‘∏’∑” „Àâ arctan (3) =
p p 3 ·≈– ®–‰¥â tan = ....................................................... 2 2 p p ‡π◊ËÕß®“° tan 0 ·≈– 2 2 ..........................................................................................................
p 0 .......................................................................................................... 2
¥—ßπ—Èπ
Y
À“ sin ‚¥¬æ‘®“√≥“®“°√Ÿª 3 sin = ....................................................... 10 3 sin arctan (3) = .......................................................................................................... 10
¥—ßπ—Èπ
6.
7.
[
®ßÀ“§à“µàÕ‰ªπ’È 1 1) sin [arcsin ] 2
= p
4) arccos 0
=
6) arctan (1)
=
8) arcsec 2
=
10) arccot 0
=
12) arccsc (2)
=
[
..............................
p = cos ( ) 4 2 = 2
2) arccos (1)
1 2) cos arccos ( ) 2
Ê pˆ = sin Á ˜ Ë 6¯ 1 = 2
[
1
]
®ßÀ“§à“À≈—°„π·µà≈–¢âÕµàÕ‰ªπ’È 0 1) arcsin 0 = .................... p 3) arctan 3 = .................... 3 p 5) arcsin (1) = .................... 2 p 7) arccot 3 = .................... 6 3p 9) arccsc ( 2) = .................... 4 3p 11) arcsec ( 2) = .................... 4
3) cos arcsin (
3
2 ) 2
]
..............................
]
..............................
= arcsin =
p ) 3
3 2
O
.................... p .................... 2 p .................... 4 p .................... 3 p .................... 2 5p .................... 6
Ê 2 pˆ = cos Á ˜ Ë 3¯ 1 = 2 4) arcsin (sin
p .............................. 3
X
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
190 p 5) arccos cos ( ) 4 2 arccos = 2 p = 4
..............................
3p 4 arcsin (1) =
..............................
=
[
8.
]
[
6) arcsin tan
]
.............................. p .............................. 2
®ßÀ“§à“¢Õß 3 ) 5 3 arcsin 5
2 2) sin arccos ( ) 3 2 arccos ( ) = 3
[
1) cos (arcsin
«‘∏’∑” „Àâ
=
«‘∏’∑” „Àâ
2 ........... 3
3 ........... 5
sin =
]
p p 2 2
cos = 0 p Y
Y
3
5
O
5
3 4
®“°√Ÿª cos [arcsin 35 ]
= sin
4 .................... 5
=
3 3) tan arcsin ( ) 4 3 arcsin ( ) = 4
[
]
«‘∏’∑” „Àâ
5 .................... 3
1 4) sin arctan ( ) 3 1 arctan ( ) = 3
[
]
«‘∏’∑” „Àâ
3 ............... 4
sin =
X
®“°√Ÿª sin [arccos ( 23 )]
= cos =
O
2
X
p p 2 2
1 ............... 3
tan =
p p 2 2 Y
Y 7
O 4
X 3
®“°√Ÿª tan [arcsin ( 43 )]
O
3
X 1
10
®“°√Ÿª sin [arctan ( 13 )] ......................................................................
191
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 = sin
= tan
......................................................................
1 = 10
3 ................................................................... 7
=
3 ) 5 3 arccos = 5 3 cos = 5 4 sin = 5
...................................................................... 1 3 1 arctan = 3 1 tan = 3
[
5) sin (2 arccos
6) sin 2 arctan
«‘∏’∑” „Àâ
«‘∏’∑” „Àâ
............... ...............
3 ) 5
sin (2 arccos
]
...............
sin (2 arctan =
1 ) 3
sin 2
2 tan
= sin 2
=
2 sin cos = ................................... 4 3 2 = ................................... 5 5 24 = ................................... 25
Ê 1ˆ 2Á ˜ Ë 3¯ = ................................... 2 Ê 1ˆ 1 Á ˜ Ë 3¯
2
1 tan
2 3 = ................................... 10 3
= 1................................... 5 12 ) 13 12 arcsin = 13 12 sin = 13 12 cos (2 arcsin ) 13
7) cos (2 arcsin
8) cos (2 arctan
3)
«‘∏’∑” „Àâ
«‘∏’∑” „Àâ
3 =
arctan
tan =
....................
= cos 2
cos (2 arctan =
3)
cos 2 2
2
= 12 sin
=
2
Ê 12ˆ 1 2 Á ˜ = ................................... Ë 13¯
=
Ê 144ˆ 1 2 Á = ................................... ˜ Ë 169¯
=
288 = ................................... 1 169 119 = ................................... 169
=
1 tan 2
1 tan 2 1 ( 3) ..... .............................. 2 1 ( 3)
1 .............................. 3 ..... 1 3 .....1.............................. 2
3 ....................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
192 9.
®ßÀ“§à“¢Õß 1) sin (arcsin
12 4 arcsin ) 13 5 Y
Y
13
5
12
«‘∏’∑” „Àâ arcsin
O
5
12 13
=
sin =
·≈–
12 13
4
X
O
4 5
arcsin
X
3
= 4 5
sin =
sin ( ) = sin cos cos sin
12 3 5 4 = ............................................. 13 5 13 5 36 20 = ............................................. 65 65 56 = ............................................. 65 2) cos (arctan
15 7 arcsin ) 8 25 Y Y 17 15
25
O
«‘∏’∑”
X
8
„Àâ arctan
15 8
=
tan =
·≈–
15 .......... 8
7
O
arcsin
24
7 25
=
sin =
cos ( ) = cos cos sin sin 8 24 15 7 = ............................................. 17 25 17 25 192 105 = ............................................. 425 425 297 = ............................................. 425
7 .......... 25
X
193
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
3) cos (arcsin
77 3 arcsin ) 85 5 Y Y 85 77
5
O
«‘∏’∑” „Àâ arcsin
36
77 85
=
sin = cos ( ) =
·≈–
3
X
O
arcsin
77 ..........
3 5
X
4
=
sin =
85 cos cos sin sin
=
.................................................. 36 4 77 3 ..................................................
=
..................................................
=
..................................................
=
..................................................
3 .......... 5
85 5 85 5 144 231 425 425 375 425 15 17
5 1 arctan ) 13 2 12 «‘∏’∑” „Àâ arcsin 13 = 5 sin = , p2 p2 ..................................................................................................................... 13 1 arctan = 2 1 tan = , p2 p2 ..................................................................................................................... 2 4) sin (arcsin
1 5 arctan ) 2 13 sin (
) = .............................................................................................. sin cos cos sin = .............................................................................................. sin (arcsin
5 2 12 1 = .............................................................................................. 13 5 13 5 10 12 = .............................................................................................. 13 5
22 5 = .............................................................................................. 65
13
5
12 5
1
2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
194
5) cos (arcsin
1 4 arccos ) 5 2
«‘∏’∑” „Àâ arcsin
1 2
=
sin = arccos
4 5
1
1 2
1
...................................................
=
cos = cos (arcsin
2
...................................................
4 ................................................... 5
4 3
5
4 1 arccos ) 5 2
= cos cos sin sin .............................................................................................. 1 4 1 3 = .............................................................................................. 2 5 2 5
43 = .............................................................................................. 5 2 2 = .............................................................................................. 10
6) sin (arccos aarcsin b)
«‘∏’∑”
„Àâ arccos a
= A
1
a cos A = .......... 0 A p arcsin b = B
1 a
A a
p p b sin B = .......... B 2 2
1
b
sin (arccos aarcsin b) = sin (AB) sin A cos B cos A sin B = .................................................................................... 2 2 1 a 1 b a b = ....................................................................................
2 2 (1 a )(1 b ) ab = ....................................................................................
1 3 arccos ) 2 5 „Àâ arcsin 12 = 1 sin = ............................................. , p2 p2 2 3 arccos = 5
7) sin (arcsin
«‘∏’∑”
B 1 b
2
2
195
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 3 0 p cos = ............................................. 5 1 3 sin (arcsin arccos ) 2 5
,
2
= sin ( )
3
sin cos cos sin = ...................................................................................
3 4 1 3 = ................................................................................... 2 5 2 5
5 4
34 3 = ................................................................................... 10 10.
3
®ß· ¥ß«à“ 2
1) cos (2 arcsin x) = 12x
«‘∏’∑”
„Àâ
arcsin x =
·≈– p2
sin = x
p 2
cos (2 arcsin x) = cos 2 2
sin = 12 .................................................. 2
= 12x .................................................. 2) sin (2 arccos x) = 2 x 1 x
«‘∏’∑”
„Àâ
2
arccos x = cos = x
·≈– 0 p
sin (2 arccos x) = sin 2 2 sin cos = .................................................. 2 1x 2 x = .................................................. 2
2 x 1x = .................................................. 3) sin (2 arcsin x) = 2 x 1 x
«‘∏’∑”
„Àâ
2
arcsin x =
·≈– p2
sin = x
p 2
sin (2 arcsin x) = sin 2 2 sin cos = .................................................. 2
2 x 1x = .................................................. 4) arctan xarctan (x) = 0
«‘∏’∑”
1
„Àâ
arctan x =
,
tan = x
p p 2 2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
196
arctan (x) =
,
p p 2 2
tan = x tan tan tan ( ) = .................................................. 1 tan tan x(x ) = .................................................. 1x(x ) 0 = ..................................................
= 0
¥—ßπ—Èπ
5) arcsin xarccos x =
«‘∏’∑”
„Àâ
p 2
arcsin x =
,
p p 2 2
sin = x arccos = cos = x 0 p
,
sin ( ) = sin cos cos sin = xx 1 x 2
2
1 x
2
2
x (1x ) = .................................................. 1 = .................................................. p sin = .................................................. 2 p = .................................................. 2
¥—ßπ—Èπ arcsin xarccos x 3 24 = arctan 4 7 3 „Àâ arctan 4
=
p 2
6) 2 arctan
«‘∏’∑”
=
,
tan = tan (2 arctan
p p 2 2
3 4
3 ) = tan 2 4 2 tan = 2 1 tan 3 2Ê ˆ Ë 4¯ = .................................................. 2 3 1 Ê ˆ Ë 4¯
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
197
3 2 = ...................................................................................................................................................... 7 16
3 16 = ...................................................................................................................................................... 2 7 24 = ...................................................................................................................................................... 7 3 24 2 arctan = arctan ...................................................................................................................................................... 4 7
¥—ßπ—Èπ
11.
®ßÀ“§à“¢Õß arcsin 45 arcsin
12 16 arcsin 13 65 4 p p arcsin = A A 5 2 2
«‘∏’∑” „Àâ
,
sin A =
®–‰¥â
arcsin
12 13
= B
,
sin B = arcsin
16 65
4 3 .......... ·≈– cos A = .......... 5 5
12 5 .......... ·≈– cos B = .......... 13 13
= C
sin C =
p p B 2 2
,
p p C 2 2
16 63 .......... ·≈– cos C = .......... 65 65
= sin (ABC) = sin (AB)C
[
]
= sin (AB) cos Ccos (AB) sin C = (sin A cos Bcos A sin B) cos C(cos A cos Bsin A sin B) sin C 3 5 4 12 16 Ê 4 5 3 12ˆ 63 = Á ˜ Ê ˆ Ë 5 13 5 13¯ 65 Ë 5 13 5 13¯ 65
..................................................
=
56 63 Ê 33ˆ 16 .................................................. 65 65 Ë 65¯ 65
=
3528528 ..................................................
4225 3000 = .................................................. 4225 120 = .................................................. 169
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
198
12.
xy 1 xy
®ßÀ“§à“¢Õß arctan xarctan yarctan «‘∏’∑” „Àâ
arctan x = A
x ·≈– p ®–‰¥â tan A = .......... 2
„Àâ
arctan y = B
y ·≈– ®–‰¥â tan B = ..........
‡¡◊ËÕ p2
A
p 2
·≈– p2
B
p 2 p p B 2 2
A
p 2
p AB .......... p ®–‰¥â .......... p ·µà arctan xarctan y p2 ®–‰¥â AB .......... 2
π—Ëπ§◊Õ
p AB 2
p ..........
·∑π§à“ ®–‰¥â
tan (AB) =
tan A tan B 1 tan A tan B
tan (AB) =
xy ........................................ 1 xy
AB =
arctan ........................................ 1 xy
arctan xarctan yarctan
xy 1 xy
=
xy
0 ........................................
2.9 ‡Õ°≈—°…≥å·≈– ¡°“√µ√’ ‚°≥¡‘µ‘ 2.9.1 ‡Õ°≈—°…≥å ‡Õ°≈—°…≥åæ◊Èπ∞“π sin = cos = tan = tan = cot =
1 csc 1 sec 1 cot sin cos cos sin
2
2
sin cos = 1 2
2
2
2
sec tan = 1 csc cot = 1
199
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
tan x csc x
µ—«Õ¬à“ß∑’Ë 1 ®ßæ‘ Ÿ®πå«à“ 2
tan x csc x
«‘∏’∑”
= cot x
2
1 tan x
2
1 tan x
=
=
sin x 1 2 cos x sin x 2
sec x
tan x =
1 sin x cos x 1
csc x = 2
cos x sin x
2
=
2
1tan x = sec x
2
cos x =
sin x cos x 1 sin x
cos x sin x cos x cos x sin x
= cot x
= cot x
µ—«Õ¬à“ß∑’Ë 2 ®ßæ‘ Ÿ®πå«à“ (sec xtan x)(1sin x) = cos x «‘∏’∑”
(sec xtan x)(1sin x) = =
sin x ˆ Ê 1 Á ˜ (1 sin x) Ë cos x cos x¯ (1 sin x)(1 sin x) cos x 2
1 sin x = cos x
tan x =
2
=
cos x sec x = 1
sin x cos x 2
(ab)(ab) = a b
cos x cos x
2
2
sin xcos x = 1
= cos x
°‘®°√√¡∑’Ë 2.9.1 (1) 1.
®ßæ‘ Ÿ®πå‡Õ°≈—°…≥å 1) sin cot = cos
«‘∏’∑”
sin cot = sin
cos sin
cos = ........................................
2) cos tan = sin
«‘∏’∑”
cos tan = cos
sin cos
sin = ........................................
2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
200 3) sin sec = tan
«‘∏’∑”
4) sec cot = csc
sin sec
sec cot
«‘∏’∑”
1 cos cos sin 1 = ........................................ sin
1 cos sin = cos tan = ........................................
= sin
2
2
2
2
2
tan cot sin
«‘∏’∑”
2
2
= cos sec
= 1sin
1 = ........................................
cos = ........................................
2
7) (csc 1)(csc 1) = cot
«‘∏’∑”
(csc 1)(csc 1)
2
2
2
tan cot cos
«‘∏’∑”
2
2
1cos = ........................................
2
sin = ........................................
2
cot = ........................................ 2
2
2
9) sin (1cot ) = 1 2
2
2
10) (1tan ) sin = tan
2
sin (1cot ) 2
«‘∏’∑”
2
2
2
(1tan ) sin 2
2
= sin csc
= sec sin
1 = ........................................
=
11) cos (tan cot ) = csc
«‘∏’∑”
2
8) tan cot cos = sin
= csc 1
«‘∏’∑”
2
6) tan cot sin = cos
cos (1tan ) 2
csc = ........................................ 2
5) cos (1tan ) = 1
«‘∏’∑”
=
cos (tan cot ) =
Ê sin cos ˆ cos Á ˜ Ë cos sin ¯
2 1 2 sin cos 2 tan = ........................................ 2
2
«‘∏’∑”
2
cos sin 2
sin cos sin 1 = ........................................ sin =
csc = ........................................
2
2
2
cot cos 2
=
2
= sin
2
12) cot cos = cot cos
=
cos 2
sin
2
cos
Ê 1 ˆ 2 Á 2 1˜ cos Ë sin ¯ 2
2
= (csc 1) cos 2
2
cot cos = ........................................
201
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 13) sec cos = sin tan
«‘∏’∑”
14) csc sin = cot cos
sec cos
«‘∏’∑”
=
1 cos cos
=
1 cos cos
=
sin cos
=
1 sin = ........................................ sin
2
2
cos = ........................................ sin cos = ........................................ cos sin
sin cos
sin tan = ........................................ 2
2
2
cot cos = ........................................
2
2
15) sec csc = sec csc
«‘∏’∑”
2
1 sin sin 2
2
= sin
csc sin
2
sec csc
«‘∏’∑”
2
2
2
2
2
2
(sin 1)(cot 1) 2
= sec (1cot ) 2
2
16) (sin 1)(cot 1) = 1csc 2
= (sin 1)csc 2
2
2
2
sin csc csc = ........................................
sec sec cot = ........................................ 2
2 1 cos sec 2 = ........................................ 2 cos sin
2
1csc = ........................................
2 1 sec 2 = ........................................ sin 2
2
sec csc = ........................................ 2
17)
2
sin cos
2
= sec
2
cos 2
«‘∏’∑”
18)
2 sec 2 sin = tan csc
2
sin cos
«‘∏’∑”
2
cos
1
=
=
2
cos 2 sec = ........................................ 4
19)
4
4
20)
sin x = csc cot 1 cos
4
sin cos sin cos 2
«‘∏’∑” 2
2
2 sec 2 csc csc sin
tan = ........................................
sin cos = sin cos sin cos
«‘∏’∑”
2 sec 2 sin csc
sin x 1 cos
2
=
(sin cos )(sin cos ) (sin cos )
=
=
(sin cos )(sin cos ) 1 sin cos
=
sin x 1 cos 1 cos 1 cos sin (1 cos ) 2
1 cos
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
202 cos = sin ........................................
=
sin (1 cos ) sin 2
=
1 cos sin
=
1 cos sin sin
cot = csc ........................................ 21)
tan 1 1 cot = tan 1 1 cot tan 1 tan 1
22)
«‘∏’∑”
=
1 1 cot 1 1 cot
=
1 cot cot 1 cot cot
cot 1 tan 1 = cot 1 tan 1 cot 1 cot 1
«‘∏’∑”
2 1 sin = (sec tan ) 1 sin 1 sin 1 sin 1 sin 1 sin = 1 sin 1 sin
24)
«‘∏’∑”
=
(1 sin )
=
1 tan tan 1 tan tan
1 tan = ........................................ 1 tan
cot = 1........................................ 1 cot 23)
=
1 1 tan 1 1 tan
1 sec = csc sin tan 1 sec sin tan 1 sec = sin sin sec
«‘∏’∑”
2
1 sec = ........................................ sin (1 sec )
2
1 sin 2
(1 sin ) = ........................................ 2 cos
1 = ........................................ sin
csc = ........................................
2
Ê 1 sin ˆ = ........................................ Á ˜ Ë cos cos ¯ 2
(sec tan) = ........................................ 2
25)
tan cot sec = 2 tan cot tan 1
«‘∏’∑”
tan cot tan cot
26)
sin
4
2
csc (1 cot )
«‘∏’∑”
= sin
sin 2
csc (1 cot )
203
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 1 tan 1 tan tan tan
=
=
2
2
sin = ........................................ 2 csc
tan 1 2 tan 1
=
2
sec = ........................................ 2 tan 1 27)
(1 cos ) 2
sin
«‘∏’∑”
2
sec 1 sec 1
=
(1 cos )
4
sin = ........................................ 2
28) (sec tan ) =
1 sin 1 sin
2
«‘∏’∑”
2
sin =
sin 2 1 (csc ) sin
(1 cos )(1 cos )
(sec tan ) =
2
1 cos
(1 cos )(1 cos ) = ........................................ (1 cos )(1 cos )
=
2
Ê 1 sin ˆ Á ˜ Ë cos cos ¯
(1 sin )
2
2
2
cos (1 sin )(1 sin ) = ........................................ 2 1 sin
(1 cos ) = ........................................ (1 cos )
(1 sin )(1 sin ) = ........................................ (1 sin )(1 sin )
1 1 sec = ........................................ 1 1 sec
(1 sin ) = ........................................ (1 sin )
sec 1 = ........................................ sec 1
sec x tan x sec x tan x
2
29) (sec x tan x) =
«‘∏’∑”
2
(sec x tan x) =
sin x ˆ Ê 1 Á ˜ Ë cos x cos x¯
=
Ê 1 sin xˆ Á ˜ Ë cos x ¯
= = =
2
30) (csc xcot x) =
«‘∏’∑” 2
2
csc x cot x csc x cot x 2
(csc xcot x)
2
cos x ˆ Ê 1 = ........................................ Á ˜ Ë sin x sin x ¯ 2
Ê 1 cos x ˆ = ........................................ ˜ Á Ë sin x ¯ 2
2
(1 cos x ) = ........................................ 2 sin x
2
1 sin x
(1 cos x ) = ........................................ 2 1 cos x
(1 sin x)(1 sin x) (1 sin x)(1 sin x)
(1 cos x )(1 cos x ) = ........................................ (1 cos x )(1 cos x )
(1 sin x) 2
cos x (1 sin x) 2
2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
204
=
1 sin x 1 sin x
1 cos x = ........................................ 1 cos x
=
1 sin x cos x 1 sin x cos x
1 cos x sin x = ........................................ 1 cos x sin x
=
1 sin x cos x cos x 1 sin x cos x cos x
1 cos x sin x sin x = ........................................ 1 cos x sin x sin x csc x cot x = ........................................ csc x cot x
sec x tan x = ........................................ sec x tan x 31) (tan x1) cos x = sin xcos x
«‘∏’∑”
(tan x1) cos x =
32) (csc xcot x)(sec x1) = tan x
«‘∏’∑”
Ê sin x ˆ 1˜ cos x Á Ë cos x ¯
.............................................. (csc xcot x)(sec x1) cos x ˆ Ê 1 ˆ Ê 1 = ........................................ 1˜ ˜Á Á ¯ Ë sin x sin x ¯ Ë cosx
sin x = ........................................ cos xcos x cos x
= ........................................ Ê 1 cos x ˆ Ê 1 cos x ˆ ˜ ˜Á Á Ë sin x ¯ Ë cos x ¯
xcos x = sin ........................................
2
1 cos x = ........................................ sin x cos x 2
sin x = ........................................ sin x cos x sin x = ........................................ cos x = ........................................ tan 33)
1 1 = 2tan x sec x 1 sin x 1 sin x
«‘∏’∑”
1 1 1 sin x 1 sin x (1 sin x) (1 sin x) = (1 sin x)(1 sin x) 2 sin x = ........................................ 2 1 sin x 2 sin x = ........................................ 2 cos x
2 sin x 1 = ........................................ cos x cos x = ........................................ 2 tan x sec x
4
2
4
34) sec xtan x =
«‘∏’∑”
4
1 sin x 2
cos x 4
sec xtan x 2
2
2
2
= (sec xtan x)(sec xtan x) 2 Ê 1 sin x ˆ = ........................................ 1 Á 2 ˜ 2 Ë cos x cos x ¯
2
1 sin x = ........................................ 2 cos x
205
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
35) (1sin x)(sec xtan x) = cos x
cos x = 1sin x 1 sin x
36)
2
«‘∏’∑”
«‘∏’∑”
(1sin x)(sec xtan x)
cos x 1 sin x
sin x ˆ Ê 1 (1 sin x )Á ˜ = ........................................ Ë cos cos x ¯
=
Ê 1 sin x ˆ (1 sin x )Á ˜ = ........................................ Ë cos x ¯
(1 sin x )(1 sin x ) = ........................................ 1 sin x
2
2
1 sin x = ........................................ cos x
1 sin x 1 sin x
1sin x = ........................................
2
cos x = ........................................ cos x cos x = ........................................ 4
4
2
37) cos xsin x = 12 sin x
«‘∏’∑”
4
4
2
2
2
2
cos xsin x = (cos xsin x)(cos xsin x) 2
2
((1sin x)sin x)1 = ................................................................................ 2
12 sin x = ................................................................................ 38)
sin x 1 cos x = 2 csc x 1 cos x sin x
«‘∏’∑”
sin x 1 cos x 1 cos x sin x
2
2
sin x (1 cos x) (1 cos x) sin x 2 2 ( 1 cos x ) (1 2 cos x cos x ) = ................................................................................ (1 cos x ) sin x
=
2 2 cos x = ................................................................................ (1 cos x ) sin x 2 (1 cos x ) = ................................................................................ 1 (1 cos x ) csc x 2 csc x = ................................................................................ 39)
csc x = cos x tan x cot x
«‘∏’∑”
csc x tan x cot x
=
csc x sin x cos x cos x sin x
csc x = ................................................................................ 2 2 sin x cos x sin x cos x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
206
csc x = ................................................................................ 1 sin x cos x
csc x sin x cos x = ................................................................................ 1cos x = ................................................................................ cos x = ................................................................................ 3
40)
3
2 2 sec x cos x = 1cos xsec x sec x cos x 3
«‘∏’∑”
3
sec x cos x sec x cos x
2
=
2
(sec x cos x)(sec x sec x cos x cos x) (sec x cos x) 2
2
sec x1cos x = ................................................................................ 2
2
1cos xsec x = ................................................................................ 2
41)
cos x 2
1 sin x cos x
1 sin x 2 sin x
= 2
«‘∏’∑”
2
cos x 1 sin x ........................................ = ...................................................... 2 2 2 sin x sin x 1 sin x (1 sin x ) (1 sin x )(1 sin x ) = ...................................................... (1 sin x )(2 sin x ) 1 sin x = ...................................................... 2 sin x 2
2
42) (1tan x) = sec x(12 cos x sin x) 2 2 (1tan x) = ...................................................... 12 tan x tan x ...................................
«‘∏’∑”
2
sin x sin x 1 2 = ...................................................... cos x cos 2 x 2
2
cos x 2 sin x cos x sin x = ...................................................... 2 cos x 2
2
(sin x cos x ) 2 sin x cos x = ...................................................... 2 cos x 1 2 sin x cos x = ...................................................... 2 cos x 2
sec x(12 sin x cos x) = ......................................................
207
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
2
43) (1cot x) = csc x (12 cos x sin x)
«‘∏’∑”
(1cot x)
2
2
12 cot xcot x = ................................................................................ 2
2 cos x cos x 1 = ................................................................................ 2 sin x sin x 2
2
sin x 2 cos x sin x cos x = ................................................................................ 2 sin x 2
2
(sin x cos x ) 2 cos x sin x = ................................................................................ 2 sin x
1 2 cos x sin x = ................................................................................ 2 sin x 2
csc x (12 cos x sin x) = ................................................................................ 44)
2
1 sin x Ê cos x ˆ = Á ˜ Ë 1 sin x¯ 1 sin x
Ê cos x ˆ Á ˜ Ë 1 sin x¯
«‘∏’∑”
2
2
cos x
=
(1 sin x)
2
2
1 sin x
=
(1 sin x)
2
(1 sin x )(1 sin x ) = ................................................................................ (1 sin x )(1 sin x ) 1 sin x = ................................................................................ 1 sin x
45)
2 1 cos x = (csc xcot x) 1 cos x
«‘∏’∑”
(csc xcot x)
2
Ê 1 cos xˆ Ë sin x sin x ¯
=
Ê 1 cos xˆ Á ˜ Ë sin x ¯ (1 cos x) 2
tan x = csc x sec x cos x
«‘∏’∑”
=
=
46)
2
2
2
sin x 2 (1 cos x ) = ........................................ 2 1 cos x
tan x ............................................... sec x cos x
tan x = ........................................ 1 cos x cos x tan x = ........................................ 2 1 cos x cos x
tan x cos x = ........................................ 2 1 cos x sin x = ........................................ 2 sin x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
208 (1 cos x )(1 cos x ) = ........................................ (1 cos x )(1 cos x )
1 = ........................................ sin x csc x = ........................................
1 cos x = ........................................ 1 cos x 47) sec x csc x2 cos x csc x = tan xcot x
«‘∏’∑”
sec x csc x2 cos x csc x =
1 2 cos x cos x sin x sin x 2
=
1 2 cos x cos x sin x
=
(1 cos x) cos x cos x sin x
2
2
2
2
sin x cos x = .......................................................... cos x sin x 2
2
sin x cos x = .......................................................... cos x sin x cos x sin x sin x cos x = .......................................................... cos x sin x tan xcot x = .......................................................... 48)
sin x tan x = sin x cos x 1 tan x
«‘∏’∑”
sin x sin x cos x
=
sin x cos x sin x cos x cos x
tan x = .......................................................... sin x cos x cos x cos x tan x = .......................................................... tan x 1
‡Õ°≈—°…≥åº≈∫«°·≈–º≈µà“ߢÕß®”π«π®√‘ßÀ√◊Õ¡ÿ¡ sin ( ) = sin cos cos sin cos ( ) = cos cos sin sin tan ( ) =
tan tan 1 tan tan
209
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
sin 2 = 2 sin cos 2
2
cos 2 = cos sin 2
= 2 cos 1 2
= 12 sin tan 2 =
2 tan 2
1 tan
sin 2 =
2 tan 2 1 tan
cos 2 =
2 1 tan 2 1 tan
ˆ ˆ cos Ê Ë 2 ¯ 2 ¯
sin sin =
2 sin Ê Ë
sin sin =
2 cos Ê Ë
ˆ ˆ sin Ê ¯ Ë 2 2 ¯
cos cos =
2 cos Ê Ë
ˆ ˆ cos Ê Ë 2 ¯ 2 ¯
cos cos =
2 sin Ê Ë
ˆ ˆ sin Ê ¯ Ë 2 2 ¯
µ—«Õ¬à“ß°“√æ‘ Ÿ®πå‡Õ°≈—°…≥å 1. 1sin = (cos
«’∏’∑”
2 sin ) 2 2
1sin
2.
cos sin = sec 2tan 2 cos sin sin «‘∏’∑” cos cos sin
2 2 = Ê cos sin ˆ sin Ë 2 2¯
=
㪉 cos2 Asin2 A = 1)
=
(
2 2 = Ê cos sin ˆ 2 sin cos Ë 2 2¯ 2 2
(
2
=
2
2
cos sin
1 2 cos sin cos 2 2
(cos Asin A = 1
ˆ 2 2¯
[(ab)2 = a22abb2]
2
cos 2 cos sin sin
2
㪉 sin 2A = 2 sin A cos A)
= Ê cos sin Ë 2
(cos sin )(cos sin ) (cos sin )(cos sin )
2
2
cos 2 = cos sin )
1sin 2 (sin 2A = 2 sin A cos A) cos 2 1 sin 2 = cos 2 cos 2 =
= sec 2tan 2
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
210 1 sin 2 1 sin 2 1 1
3.
«‘∏’∑”
cos cos sin 2 sin 2
2 = cot 2 cos 2 cos 2
4.
cos 3 cos = cot sin 3 sin cos 3 cos sin 3 sin
«‘∏’∑”
3 ˆ 3 ˆ cos Ê Ë 2 ¯ 2 ¯ = 3 ˆ 3 ˆ 2 cos Ê sin Ê Ë 2 ¯ Ë 2 ¯ 2 cos Ê Ë
(1 cos 2 ) sin 2 = (1 cos 2 ) sin 2 2
2 cos 2 cos 2 cos 2 sin
2 cos 2 sin cos 2 2 sin 2 sin cos
=
[cos 2A = 2 cos2 A1
= cot
=
2
cos 2A = 12 sin A sin 2A = 2 sin A cos A
]
2 cos (cos sin ) 2 sin (cos sin )
=
= cot
°‘®°√√¡∑’Ë 2.9.1 (2) 1.
®ßæ‘ Ÿ®πå‡Õ°≈—°…≥å cos 3 cos = cot 2 sin 3 sin
1)
«‘∏’∑”
cos 3 cos sin 3 sin
2 cos Ê 3 ˆ cos Ê 3 ˆ Ë 2 ¯ Ë 2 ¯ = 3 ˆ 3 ˆ 2 sin Ê cos Ê Ë 2 ¯ Ë 2 ¯ =
cos 2 sin 2
cot 2 = ........................................
2)
«‘∏’∑”
sin 2 sin 2 = tan ( ) cos 2 cos 2 sin 2 sin 2 cos 2 cos 2
=
=
2 sin Ê 2 2 ˆ cos Ê 2 2 ˆ Ë 2 ¯ Ë 2 ¯ 2 cos Ê Ë
2 2 ˆ 2 2 ˆ cos Ê 2 ¯ Ë 2 ¯
sin ( ) cos ( )
tan ( ) = ........................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
3)
sin x sin y Ê x yˆ = tanÁ ˜ Ë 2 ¯ cos x cos y
«‘∏’∑”
sin x sin y cos x cos y
x yˆ x yˆ 2 sin Ê cos Ê Ë 2 ¯ Ë 2 ¯ = ................................................................................ x yˆ x yˆ cos Ê 2 cos Ê Ë 2 ¯ Ë 2 ¯ Ê x yˆ sin Á ˜ Ë 2 ¯ = ................................................................................ Ê x yˆ cos Á ˜ Ë 2 ¯
Ê x yˆ tan Á ˜ Ë 2 ¯ = ................................................................................ 4)
sin 6x sin 4 x = tan x cos 6x cos 4 x Ê 6x 4xˆ Ê 6x 4xˆ 2 cos Á ˜ sin Á ˜ Ë Ë 2 ¯ 2 ¯ sin 6x sin 4 x = ................................................................................ cos 6x cos 4 x Ê 6x 4xˆ Ê 6x 4xˆ 2 cos Á ˜ cos Á ˜ Ë Ë 2 ¯ 2 ¯
«‘∏’∑”
sin x = ................................................................................ cos x
tan x = ................................................................................ 5) cos 7xcos 5x2 cos x cos 2x = 4 cos 4x cos 2x cos x
«‘∏’∑”
cos 7xcos 5x2 cos x cos 2x =
2 cos Ê Ë
7x 5xˆ 7x 5xˆ cos Ê 2 cos x cos 2 x ¯ Ë ¯ 2 2
= 2 cos 6x cos x2 cos x cos 2x = 2 cos x cos 6xcos 2x
[
]
6x 2xˆ 6x 2xˆ ˘ È 2 cos x Í2 cos Ê cos Ê = .................................................................................................... Ë ¯ Ë ¯ ˙˚ 2 2 Î
[
]
2 cos x 2 cos 4x cos 2x = .................................................................................................... 4 cos 4x cos 2x cos x = .................................................................................................... 6)
cos 2 x cos 4 x = tan x sin 2 x sin 4 x
«‘∏’∑”
cos 2 x cos 4 x sin 2 x sin 4 x
=
=
Ê 2 x 4xˆ Ê 2 x 4xˆ 2 sin Á ˜ sin Á ˜ Ë 2 ¯ Ë 2 ¯ Ê 2 x 4xˆ Ê 2 x 4xˆ 2 sin Á ˜ ˜ cos Á Ë 2 ¯ Ë 2 ¯
sin (x) cos (x)
211
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
212
(sin x) cos x tan x = ................................................................................
=
7)
sin x sin 3x = tan 2x cos x cos 3x
«‘∏’∑”
8)
sin x sin 3x = cos x cos 3x
Ê x 3xˆ Ê x 3xˆ 2 sin Á ˜ cos Á ˜ Ë 2 ¯ Ë 2 ¯ Ê x 3xˆ Ê x 3xˆ 2 cos Á ˜ ˜ cos Á Ë 2 ¯ Ë 2 ¯
=
sin 2 x ................................................................................ cos 2 x
=
tan 2x ................................................................................
sin (x y) sin (x y) = tan x cot y sin (x y) sin (x y)
«‘∏’∑”
1 {sin (x y) sin (x y)} 2 1 {sin (x y) sin (x y)} 2 sin x cos y = ................................................................ cos x sin y
sin (x y) sin (x y) = sin (x y) sin (x y)
tan x cot y = ................................................................ 9)
cos 5x 2 cos 3x cos x = cot 3x sin 5x 2 sin 3x sin x 5x 2 cos 3x cos x «‘∏’∑” cos = sin 5x 2 sin 3x sin x
(cos 5x cos x) 2 cos 3x (sin 5x sin x) 2 sin 3x
Ê 5x xˆ Ê 5x xˆ 2 cos Á ˜ cos Á ˜ 2 cos 3 x Ë ¯ Ë 2 ¯ 2 = ............................................................................. Ê 5x xˆ Ê 5x xˆ 2 cos Á ˜ sin Á ˜ 2 sin 3 x Ë 2 ¯ Ë 2 ¯
2 cos 3 x cos 2 x 2 cos 3 x = ............................................................................. 2 cos 3 x sin 2 x 2 sin 3 x 2 cos 3 x (cos 2 x 1) = ............................................................................. 2 sin 3 x (cos 2 x 1)
cos 3 x = ............................................................................. sin 3 x = cot ............................................................................. 3x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
10)
sin 3 cos 3 = 2 sin cos sin 3 cos 3 = sin cos
sin 3 cos cos 3 sin sin cos sin (3 ) = ................................................................................ sin cos sin 2 = ................................................................................ sin cos
«‘∏’∑”
2 sin cos = ................................................................................ sin cos 2 = ................................................................................ 11)
cos 3 cos = tan 2 sin 3 sin
«‘∏’∑”
cos 3 cos sin 3 sin
=
Ê 3 ˆ Ê 3 ˆ 2 sin Á ˜ sin Á ˜ Ë 2 ¯ Ë 2 ¯ Ê 3 ˆ Ê 3 ˆ 2 cos Á ˜ sin Á ˜ Ë 2 ¯ Ë 2 ¯
2 sin 2 sin = ................................................................................ 2 cos 2 sin 2 = tan ................................................................................ 12)
cos 3 cos = cot 2 sin 3 sin
«‘∏’∑”
cos 3 cos sin 3 sin
Ê 3 ˆ Ê 3 ˆ 2 cos Ë ¯ cos Ë 2 ¯ 2 = ................................................................................ 2 sin Ê 3 ˆ cos Ê 3 ˆ Ë 2 ¯ Ë 2 ¯ 2 cos 2 cos = ................................................................................ 2 sin 2 cos cos 2 = ................................................................................ sin 2
cot 2 = ................................................................................ 13)
sin sin 2 sin 3 = tan 2 cos cos 2 cos 3 sin sin 2 sin 3 «‘∏’∑” cos = cos 2 cos 3
(sin 3 sin ) sin 2 (cos 3 cos ) cos 2 2 sin 2 cos sin 2 = ................................................................................ 2 cos 2 cos cos 2 sin 2 (2 cos 1) = ................................................................................ cos 2 (2 cos 1)
2 = tan ................................................................................
213
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
214
14)
1 sin 1 sin 1 1
«‘∏’∑”
2 2 sin sin
cos 2 = tan cos 2 2 cos 2 = 2 cos 2
(1 cos 2 ) sin 2 (1 cos 2 ) sin 2 2 2 sin 2 sin cos = ................................................................................ 2 2 cos 2 sin cos 2 sin (sin cos ) = ................................................................................ 2 cos (sin cos )
= tan ................................................................................ 2.
®ßæ‘ Ÿ®πå‡Õ°≈—°…≥å 2
1) (sin cos ) = 1sin 2 (sin cos )
«‘∏’∑”
2
2
2
= sin 2 sin cos cos 2
2
= (sin cos )2 sin cos 1sin 2 = ..................................................................... 2
2) sin 4 = 4 cos sin (12 sin ) sin 4 = 2 sin 2 cos 2
«‘∏’∑”
2
= 2(2 sin cos )(12 sin ) 2
sin cos (12 sin ) = 4..................................................................... 4
2
3) cos 4 = 8 cos 8 cos 1
«‘∏’∑”
2
cos 4 = 2 cos 21 2
2
2(2 cos 1) 1 = ..................................................................... 4
2
2(4 cos 4 cos 1)1 = ..................................................................... 4
2
8 cos 8 cos 1 = ..................................................................... 4)
1 tan x 1 sin 2 x = 1 tan x cos 2 x
«‘∏’∑”
1 tan x 1 tan x 1 tan x = 1 tan x 1 tan x 1 tan x 2
=
1 2 tan x tan x 2
1 tan x 2
1 tan x tan 2x 1 tan x 1 sin 2 x = cos 2 x cos 2 x
=
1 sin 2x = ..................................................................... cos 2x
215
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 5)
2
2
tan xcos 2x = 1cos 2x tan x
«‘∏’∑”
2
2
2
tan xcos 2x = tan x(12 sin x) 2
2
= 1(2 sin xtan x) 2 Ê 2 sin x ˆ = 1 Á 2 sin x 2 ˜ cos x¯ Ë
Ê 2 cos 2 x sin 2 x sin 2 xˆ = 1 Á ˜ 2 cos x Ë ¯ 2
2
= 1(2 cos x1)
sin x 2
cos x
2
1cos 2x tan x = ............................................................ 6)
2 2 tan x = 1tan x tan 2 x
2 tan x tan 2 x
«‘∏’∑”
=
2 tan x 2 tan x 2
1 tan x 2
1 tan x 2 tan x 2 x 1tan = ............................................................ = 2 tan x
7)
2 2 cos 3x sin x = cos 2xsin 2x sec x csc 3x cos 3x sin x sec x csc 3x
«‘∏’∑”
= cos 3x cos xsin 3x sin x cos (3xx) = ............................................................ cos 4x = ............................................................ 2
2
cos 2xsin 2x = ............................................................ 8)
csc xsec x 1 sin 2 x = csc xsec x cos 2 x
«‘∏’∑”
csc xsec x csc xsec x
=
1 1 sin x cos x 1 1 sin x cos x
=
cos x sin x sin x cos x cos x sin x sin x cos x
=
cos xsin x cos xsin x cos xsin x cos xsin x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
216
2
=
2
cos x 2 cos x sin x sin x 2
2
cos x sin x 2
2
(cos x sin x) 2 sin x cos x cos 2 x 1 sin 2x = ............................................................ cos 2x =
2
9) sin 4 sin 2 sin =
1 (1cos 6) 2 2
2 1 cos 6 cos 2 sin 2 2 2 1 = cos 6 (12 sin sin 2
sin 4 sin 2 sin =
«‘∏’∑”
[
]
[
]
2 2 1 1 cos 6 sin sin = ............................................................ 2 2 1 1 cos 6 = ............................................................ 2 2 1 (1cos 6 ) = ............................................................ 2 3
10)
3
sin x cos x 1 = 1 sin 2x sin x cos x 2 3
«‘∏’∑”
3
sin x cos x = sin x cos x
2
2
(sin x cos x )(sin xsin x cos x cos ) (sin x cos x)
2 2 1 = (sin xcos x) (2 sin x cos x) 2
1 1 sin 2 x = ...................................................................... 2 4
11) cos 4x = 4 cos 2x38 sin x
«‘∏’∑”
2
cos 4x = 12 sin 2x = 12(2 sin x cos x) 2
2
2
= 12(4 sin x cos x) 2
2
2
4
= 18 sin x(1sin x) = 18 sin x8 sin x 4 Ê 1 cos 2 xˆ = 1 8Á ˜ 8 sin x Ë ¯ 2 4
14(1cos 2x)8 sin x = ...................................................................... 4
144 cos 2x8 sin x = ...................................................................... 4
4 cos 2x38 sin x = ......................................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 3.
®ßæ‘ Ÿ®πå«à“ (sin 2sin )(12 cos ) = sin 3 «‘∏’∑” (sin 2sin )(12 cos ) = (2 sin cos sin )(12 cos ) 2 cos 1 cos 1) = sin (....................)(2 2 sin = ..........(4 cos 1) 2
1sin = sin 4(....................)1
[
sin 44 sin 1 = .......... 2
[
]
]
2
34 sin = sin (....................) 3 sin 4sin3 = .......... sin 3 = .................... 4.
®ßæ‘ Ÿ®πå«à“ cos 3 sin 2cos 4 sin = cos 2 sin «‘∏’∑” cos 3 sin 2cos 4 sin 1 1 sin (4)sin (4) = [sin (32)sin(32)] [.......................................... ] 2 2 1 sin 5sin 3 (sin 5sin ..............................) 2 1 sin = (sin 3...............) 2 3 ˆ ˘ 1È 3 ˆ sin Ê ............... = 2 cosÊ .......... Í 2 Ë 2 ¯ Ë ............... ¯ ˙˚ 2Î
=
cos 2 sin = ........................................ 5.
∂â“ A, B, C ‡ªìπ¢π“¥¢Õß¡ÿ¡¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ®ßæ‘ Ÿ®πå«à“ tan Atan Btan C = tan Atan Btan C
æ‘ ®Ÿ πå ‡π◊ËÕß®“° A, B, C ‡ªìπ¢π“¥¢Õß¡ÿ¡¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ¥—ßπ—Èπ ABC = 180 180 C AB = ................................................ tan(180 C) tan (AB) = ................................................ tan A tan B tan C = ................................................ t tan A tan B tan C(1tan A tan B) tan Atan B = ................................................ ................................................
tan Ctan Ctan Atan B tan Atan B = ................................................ ................................................ tan Atan Btan C tan Atan Btan C = ................................................ ................................................
217
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
218 6.
∂â“ ABC = 180 ®ßæ‘ Ÿ®πå«à“ sin Asin B sin C = cos B sin C æ‘ ®Ÿ πå ABC = 180 A = 180 (BC) sin A = sin 180 (BC)
[
]
sin A = ................................................ sin(BC) ................................................ sin A = ................................................ sin B cos Ccos B sin C ................................................ sin Asin B sin C = ................................................ cos B sin C ................................................ 7.
®ßæ‘ Ÿ®πå«à“ «‘∏’∑”
2 cos 2 cos 2 cos 3 = cot cos 2 cos 2 cos 3 2 cos 2 cos 2 cos 3 (cos 3 cos ) 2cos 2 = cos 2 cos 2 cos 3 (cos 3 cos ) 2cos 2
= =
2 cos 2 cos 2 cos 2 2 cos 2 cos 2 cos 2 2 cos 2 (cos 1) 2 cos 2 (cos 1)
Ê ˆ 2 Á 2 cos 1˜ 1 Ë ¯ 2 = ................................................ 2 ˆ Ê Á 1 2 sin ˜ 1 Ë 2¯ 2 2 cos 2 = ................................................ 2 2 sin 2 2 = cot ................................................ 2
8.
®ß· ¥ß«à“ tan A (cosec 2Acot 2A) = 1 «‘∏’∑”
tan A (cosec 2Acot 2A) = =
cos 2 A ˆ Ê 1 tan A Á ˜ Ë sin 2 A sin 2 A ¯ 1 cos 2Aˆ ..................... tan A Ê Ë sin 2 A ¯
sin A Ê 1 cos 2 A ˆ = ................................................ ˜ Á cos A Ë 2 sin A cos A¯ 1 cos 2A = ................................................ 2 2 cos A 2
1 (2 cos A 1) = ................................................ 2 2 cos A 1 = ................................................
219
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 9.
°”Àπ¥ A = BC ®ßæ‘ Ÿ®πå«à“ tan Atan Btan C = tanAtan Btan C «‘∏’∑” A = BC tan A = tan (BC) tan B tan C tan A = ............................................ ............................................ 1 tan B tan C tan A(1tan Btan C) = ............................................ tan Btan C ............................................ tan Atan Atan Btan C = ............................................ tan Btan C ............................................ tan Atan Btan C = ............................................ tan Atan Btan C ............................................
10.
®ßæ‘ Ÿ®πå«à“ cot Atan 2A = cot A sec 2A 1 2 tan A «‘∏’∑” cot Atan 2A = 2 tan A
1 tan A
2
=
(1 tan A ) 2 tan A tan A 2
tan A (1 tan A ) 2
2
1 tan A 2 tan A = .......................................................... 2 tan A (1 tan A ) 2
1 1 tan A = .......................................................... tan A 1 tan 2 A 1 1 ............................................ = tan A 1 tan 2 A 2 1 tan A
1 = cot A............................................ cos 2A = cot Asec 2A
11.
®ß· ¥ß«à“ «‘∏’∑”
o sin 12 o o = 0 sin 48 sin 81 o o o o o o sin 9 sin 12 sin 9 sin 81 sin 12 sin 48 = o o o o sin 48 sin 81 sin 48 sin 81 sin 9
o
= =
=
1 1 o o o o (cos 72 cos 90 ) .......................................... (cos 36 cos 60 ) 2 2 o o sin 48 sin 81 o o o o cos 72 cos 90 ................................. cos 36 cos 60 o o 2 sin 48 sin 81 Ê 5 1ˆ Ê 5 1ˆ 1 ˜ Á ˜ 0 Á...................... Ë 4 ¯ 2 Ë 4 ¯ o o 2 sin 48 sin 81
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
220
0 = ................................................................. o o 2 sin 48 sin 81 0 = ................................................................. 12.
®ßæ‘ Ÿ®πå«à“ tan 75 tan 30 tan 75 tan 30 = tan 45 «‘∏’∑” ®“° tan 45 = 1 tan (75 30 ) = 1 o o tan 75 tan 30 o o 1 tan 75 tan 30
= 1
1tan 75 tan 30 tan 75 tan 30 = ........................................ tan 75 tan 30 tan 75 tan 30 = ........................................ 1 ............................................................. tan 75 tan 30 tan 75 tan 30 = ........................................ tan 45 .............................................................. 13.
®ßæ‘ Ÿ®πå«à“ cos 10 sin 40 = «‘∏’∑” cos 10 sin 40 =
3 sin 70 sin 40 cos (90 80 ) ...............
= sin 80 sin 40 =
Ê 80o 40o ˆ Ê 80o 40o ˆ cos 2 sin Á ................................. Á ˜ ˜ 2 2 Ë ¯ Ë ¯
cos 20 = 2 sin 60 ...................
Ê 3ˆ o o 2 Á ˜ cos (90 70 ) = .......................................................... Ë 2¯ 3 sin 70 = ..........................................................
14.
®ßæ‘ Ÿ®πå«à“ cos 80 sin 50 cos 20 = 0 «‘∏’∑” cos 80 sin 50 cos 20 = (cos 80 cos 20 )sin 50 Ê 80o 20o ˆ Ê 80o20o ˆ o sin = 2 sin Á Á ˜ sin 50 ˜ 2 2 Ë ¯ Ë ¯
..................................................
= 2 sin 50 sin 30 sin 50
..................................................
= 2 sin 50 ÊÁ 1 ˆ˜ sin 50 Ë 2¯
..................................................
= sin 50 sin 50 =
.................................................. 0 ..................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 15.
®ßæ‘ Ÿ®πå«à“ tan 20 tan 40 tan 80 = «‘∏’∑” tan 20 tan 40 tan 80 =
3
o o o (sin 20 sin 40 ) sin 80 o o o (cos 20 cos 40 ) cos 80
o o o o 1 (cos 20 cos 60 ) sin (90 10 ) = 2 o o o 1 (cos 20 cos 60 ) cos 80 2
=
o o o (cos 20 cos 60 ) cos 10 o o o (cos 20 cos 60 ) cos 80
=
o o cos 60 cos 10 cos 20 cos 10 ............................. o cos 60 cos 80 o cos 20 cos 80 ................................
o o o o 1 1 (cos 30 cos 10 ) (cos 70 cos 50 ) ............................... 2 = 2 o o o o 1 1 (cos 100 cos 60 ) (cos 140 cos 20 ) ............................... 2 2 =
o o o o cos 30 (cos 10 cos 70 ) cos 50 o o o o (cos 100 cos 140 ) cos 60 cos 20
=
o o o o sin 60 ............................. 2 sin 30 sin 40 cos 50 o o o o 2 cos 120 cos 20 cos 60 cos 20 ...............................
=
o o o Ê 1ˆ sin 60 2Á ˜ cos 50 cos 50 Ë 2¯
[cos 50 = sin 40 ] o o o Ê 1ˆ 2 Á ˜ cos 20 cos 60 cos 20 Ë 2¯ o o o sin 60 cos 50 cos 50 = ................................................................................ o o o cos 60 cos 20 cos 20 60 = tan ................................................................................ 3 = ................................................................................
16.
p 2p 4p 8p 1 cos cos cos = 15 15 15 15 6 p 2p 4p 8p cos cos cos cos 15 15 15 15
®ßæ‘ Ÿ®πå«à“ cos «‘∏’∑”
4p p 8p 2 pˆ cos ˆ Ê cos cos = Ê cos Ë ¯ Ë 15 15 15 15 ¯
=
p p 1 2p 2 pˆ 1Ê cos cos ˆ Ê cos cos 3 5¯ 2 Ë 3 5¯ 2Ë
=
2p 2 pˆ p p Ê 1Ê cos cos cos ˆ Ë cos Ë ¯ 3 5¯ 3 5 4
..................................................
221
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
222
=
5 1ˆ 1Ê 1 5 1ˆ Ê 1 Á ˜ Á ˜ ........................................ 4 ¯ 4Ë 2 4 ¯ Ë 2
=
5 1ˆ 1Ê 1 5 1ˆ Ê 1 ˜ Á Á ˜ ........................................ 4 4¯ 4Ë 2 4 4¯ Ë 2
1Ê 5 3ˆ Ê 5 3ˆ ˜ Á ˜ ........................................ Á 4Ë 4 4¯ Ë 4 4¯ 1Ê 5 9ˆ = ........................................ Á ˜ 4Ë 16 16¯ =
1Ê 4 ˆ Á ˜ = ........................................ 4Ë 16¯ 1 = ........................................ 16
17.
∂â“
tan tan 1 = 1 tan 1 tan 2
tan tan 1 tan 1 tan
=
1 2
tan tan 1 tan tan tan tan
=
1 2
«‘∏’∑”
2 tan tan tan tan ........................................
tan tan ........................................ 1 tan tan tan ( ) ........................................ 18.
∂â“ = «‘∏’∑”
5p 4
5p 4
®ßæ‘ Ÿ®πå«à“ =
= 1tan tan tan tan 1tan tan = ........................................ 1 = ........................................
5p tan = ........................................ 4 5 p = ........................................ 4
®ßæ‘ Ÿ®πå (1tan )(1tan ) = 2 =
5p 4
= p
p 4
p tan ( ) = tan Ê p ˆ Ë 4¯ p tan tan = tan 4 1 tan tan tan tan 1 .............................................. = ........................................ 1 tan tan
1tan tan tan tan = ........................................ .............................................. 1 tan tan tan tan = ........................................ ..............................................
223
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 (1tan )tan (1tan ) = .................................................. 11 ( 1 ) ..................................................
∫«° ∑—Èß Õߢâ“ß
(1tan )(1tan ) = .................................................. 2 .................................................. 19.
®ßæ‘ Ÿ®πå«à“ (12 sin 2)(cos sin) = sin 3cos 3 «‘∏’∑” (12 sin 2)(cos sin ) (12 sin 2) sin = (12 sin 2) cos .................................................. (sin 2 sin 2 sin ) = (cos 2 sin 2 cos ).................................................. 2 sin 2 sin = cos 2 sin 2 cossin.................................................. sin[cos (2)cos(2)] = cos [sin (2)sin (2)]............................................................ sin cos 3cos = cos sin 3sin .................................................. = sin 3cos 3
20.
®ßæ‘ Ÿ®πå«à“ cos 3sin 3 = (cos sin )(14 cos sin ) «‘∏’∑” cos 3sin 3 3
3 (3 sin 4 sin ) = (4 cos 3 cos ).................................................. 3 3 3(cos sin ) = 4(cos sin ).................................................. 2
2
(cos cos sin sin ) = 4(cos sin )..................................................3(cos sin ) 2
2
[4(cos sin cos sin )3] = (cos sin )............................................................
sin )[4(1cos sin )3] = (cos ............................................................ sin )(44 cos sin 3) = (cos ............................................................ sin )(14 cos sin ) = (cos ............................................................
2.9.2 ¡°“√µ√’ ‚°≥¡‘µ‘ °“√·°â ¡°“√µ√’ ‚°≥¡‘µ‘„™â«‘∏’°“√‡¥’¬«°—∫°“√·°â ¡°“√æ’™§≥‘µ ¡°“√≈Õ°“√‘∑÷¡À√◊Õ ¡°“√‡Õ°´å‚æ‡ππ‡™’¬≈ ‚¥¬Õ“»—¬§«“¡√Ÿâ‡°’ˬ«°—∫øíß°å™—πµ√’ ‚°≥¡‘µ‘ øíß°å™π— µ√’ ‚°≥¡‘µ‰‘ ¡à‡ªìπøíß°å™π— 1-1 §à“¢Õßøíß°å™π— µ√’ ‚°≥¡‘µ¢‘ Õß®”π«π®√‘ßÀ√◊Õ¡ÿ¡„¥Ê Õ“®®–´È”°—π ∂â“‚®∑¬å ‰¡à ‰¥â°”Àπ¥„À⧔µÕ∫Õ¬Ÿà „π™à«ß„¥™à«ßÀπ÷Ëß §«√µÕ∫„π√Ÿª¢Õߧà“∑—Ë«‰ª µ—«Õ¬à“ß∑’Ë 1 ®ß·°â ¡°“√ 2 sin2sin 1 = 0 ∂â“ 0 2p 2 «‘∏’∑” 2 sin sin 1 = 0 (2 sin 1)(sin 1) = 0 2 sin 1 = 0
À√◊Õ sin 1
= 0
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
224
¥—ßπ—Èπ
1 2 7p 11p = 6 6 p 7p 11p = 2 6 6
sin =
À√◊Õ sin
,
À√◊Õ
,
= 1 p 2
=
,
µÕ∫
∂â“‚®∑¬å ‰¡à ‰¥â°”Àπ¥™à«ß¢Õß ¡“„Àâ §à“∑—Ë«‰ª§◊Õ
p 2 7p = 2np 6 11p = 2np 6
= 2np
‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡
°‘®°√√¡∑’Ë 2.9.2 2
1. tan 3 = 1 0 p
«‘∏’∑”
,
2
tan 3 = 1 tan 3 = 1 tan 3 = 1
‡¡◊ËÕ tan 3
= 1
3 =
‡¡◊ËÕ tan 3
p 5p 9p 4 4 4
,
¥—ßπ—Èπ ‡∑à“°—∫ ®ß·°â ¡°“√ «‘∏’∑”
,
, ,
,
, ,
,
= 1
3 =
,
p 5p 9p = .............................. 12 12 12 p p 5 p 7 p 9 p 11p .................................................. 12 4 12 12 12 12 2 sin x cos x = cos x 0 x 360
2.
tan 3 = 1
À√◊Õ
3p 4
,
7p 11p 4 4
,
3 p 7 p 11p , , = .............................. 12 12 12
,
2 sin x cos x = cos x
2 sin x cos xcos x = 0 2 sin x1 cos x (....................) = 0 0 cos x = ............... 90 270 x = ...............
,
À√◊Õ
0 2 sin x1 = .................... 1 sin x = .................... 2
,
30 150 x = .................... 30 , 90 , 150 , 270 ¥—ßπ—Èπ x ‡∑à“°—∫ ..................................................
225
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 3.
®ß·°â ¡°“√ 2 sin x = csc x, 0 x 360 «‘∏’∑” 2 sin x =
csc x
1 sin x 1 2 sin x = 2 1 sin x = 2
2 sin x =
....................
sin x =
1 2
....................
À√◊Õ
sin x =
1 .................... 2
x =
225 , 315 ....................
x =
45 , 135 .................... 45 , 135 , 225 , 315 ¥—ßπ—Èπ x ‡∑à“°—∫ .................................................. 4.
®ß·°â ¡°“√ 6 sin2sin 2 = 0, 0 360 2 «‘∏’∑” 6 sin sin 2 = 0 3 sin 2 2 sin 1 (....................)(....................) = 0 0 3 sin 2 = .................... 2 sin = .................... 3 0.6667 = ....................
À√◊Õ
0 2 sin 1 = .................... 1 sin = .................... 2 30 , 150 = ....................
318 11 = 221 49, ....................
, 150 , 221 49, 318 11 ¥—ßπ—Èπ ‡∑à“°—∫ 30 .................................................. 5.
®ß·°â ¡°“√ ”À√—∫ 0 x 360 1) sin (2x10 ) = 0.7660 2x10 = 50 130 360 50 360 130 = 2x = x =
, , , 410 , 490 50 , 130 , .............................. , 480 40 , 120 , 400 .............................. 220 , 240 20 , 60 , ..............................
2) cos (3x30 ) = 0.8660 3x30 = = 3x = x =
°”Àπ¥ sin 50 = 0.7660
°”Àπ¥ cos 30 = 0.8660 150 , 210 , 360 150 , 360 210 , 720 150 , 720 210 150 , 210 , 510 , 570 , 870 , 930 180 , 240 540 , 600 , 900 , 960 ...................................................................... 60 , 80 , 180 , 200 , 300 , 320 ......................................................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
226 6.
®ß·°â ¡°“√ 3 sin2 2 cos = 2, 0 360 2 «‘∏’∑” 3 sin 2 cos = 2 2 2 2 3(1cos )2 cos 2 = 0 („™â sin cos = 1) 2
33 cos 2 cos 2 = 0 2
3 cos 2 cos 1 = 0 cos 1 = 0 (3 cos 1)(...............) 3 cos 1 = 0
cos 1 = 0
À√◊Õ
...............
1
= 0.333 .................... .......... À√◊Õ 3 109 28, 250 32 = .................... .................... , 109 28, 250 32, 360 ¥—ßπ—Èπ ‡∑à“°—∫ 0 ..................................................
cos =
7.
cos =
...............
=
.................... 1 0, 360 ....................
®ß·°â ¡°“√ ‡¡◊ËÕ 0 360 1)
cos 2 = cos 1
«‘∏’∑”
cos 2 = cos 1 2
2 cos 1 = cos 1 2 cos 2cos = 0 cos (2 cos 1) = 0 0 cos = ....................
À√◊Õ
90 , 270 = ....................
0 2 cos 1 = .................... 1 cos = .................... 2 60 , 300 = ....................
¥—ßπ—Èπ ‡∑à“°—∫ 60 , 90 , 270 , 300 2)
sin cos =
3 4
«‘∏’∑”
sin cos =
3 4
1 (2 sin cos ) = 2
3 4
1 sin 2 = 2
3 4
sin 2 =
3 2
2 = 60 120 360 60 360 120 2 =
=
, , , 60 , 120 , 420 , 480 .................................................. 30 , 60 , 210 , 240 ..................................................
227
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 3)
(12 sin ) cos 2 = 0
«‘∏’∑”
(12 sin ) cos 2 = 0 12 sin = 0
cos 2 = 0
À√◊Õ
1 sin = .................... 2
cos 2 = 0
210 , 330 = ....................
90 270 450 660 2 = .....................................
=
, , , 45 , 135 , 225 , 330 .....................................
45 , 135 , 210 , 225 , 315 , 330 ¥—ßπ—Èπ ‡∑à“°—∫ ............................................................ 4)
sin = 2 cos (30 )
«‘∏’∑”
sin = 2 cos (30 ) sin = 2 cos cos 30 sin sin 30
[
]
Ê 3 ˆ 1 2Á cos sin ˜ 2 2 Ë ¯
sin =
3 cos sin sin = .................................................. 3 cos = .................................................. 0 cos = .................................................. 90 , 270 = .................................................. 5)
sin cos (30 ) = 0
«‘∏’∑”
sin cos (30 ) = 0 sin cos cos 30 sin sin 30 = 0 sin
3 1 cos sin = 0 2 2 3 1 cos sin = 0 2 2
3 cos sin = .........................
sin cos
3 = .........................
3 tan = ......................... 120 , 300 = ......................... cos 2 = cos sin
6)
«‘∏’∑”
2
2
cos sin = cos sin (cos sin )(cos sin ) = (cos sin )
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
228
cos sin cos sin (cos sin )(.........................)(.........................) = 0 cos sin 1 (cos sin )(..............................) = 0 cos sin = 0 sin = sin = cos
............... cos ............... 1 ...............
cos sin 1 = 0
À√◊Õ
cos sin = 1 1 1 cos sin = 2 2
1 ( 2 1 cos cos 45 sin sin 45 = .......... 2
tan = 1
...............
π”
1 2
§Ÿ≥∑—Èß Õߢâ“ß)
[„Àâ cos ( )]
1 cos (45 ) = .......... 2
135 , 315 = ....................
45 , 315 , 405 45 = .............................. 0 , 270 , 360 = .............................. 0 , 135 , 270 , 315 , 360 ¥—ßπ—Èπ ‡∑à“°—∫ ............................................................ 7)
«‘∏’∑”
tan 2 = 3 tan
2 tan 2
1 tan
= 3 tan 2
2 tan = 3 tan (1tan ) 2
2 tan 3 tan (1tan ) = 0 2
tan 23(1tan )
[
]
= 0
2
3 tan 1 = 0 tan (....................) 0 tan = ....................
À√◊Õ
0 , 180 , 360 = .........................
2 0 3 tan 1 = .................... 2 1 tan = .................... 3 1 1 tan = tan = 3 3
........ À√◊Õ
, 210 = 30 ........................
................
150 , 330 = .....................
0 , 30 , 150 , 180 , 210 , 330 , 360 ¥—ßπ—Èπ ‡∑à“°—∫ ................................................................. cos 3 = cos
8)
«‘∏’∑”
cos 3cos = 0 Ê 3 ˆ Ê 3 ˆ 2 sin Á ˜ sin Á ˜ = 0 Ë 2 ¯ Ë 2 ¯ 2 sin 2 sin = 0 sin 2 =
0 ..........
À√◊Õ
sin =
0 ..........
229
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 , , , , 0 , 90 , 180 , 270 , 360 = ................................................ 0 , 90 , 180 , 270 , 360 ¥—ßπ—Èπ ‡∑à“°—∫ ............................................................ 0 180 360 540 720 2 = ................................................
sin 5sin = 0
9)
«‘∏’∑”
0 , 180 , 360 = ..............................
Ê 5 ˆ Ê 5 ˆ 2 cosÁ ˜ sin Á ˜ Ë 2 ¯ Ë 2 ¯
= 0
2 cos 3 sin 2 = 0 0 cos 3 = ..................
À√◊Õ sin 2 = 0.................. 90 , 270 , 450 , 630 , 990 0 , 180 , 360 , 540 , 720 3 = ........................................................... 2 = .......................................... 30 , 90 , 150 , 210 , 270 , 330 0 , 90 , 180 , 270 , 360 = ........................................................... = .......................................... 0 , 30 , 90 , 150 , 180 , 210 , 270 , 330 , 360 ¥—ßπ—Èπ ‡∑à“°—∫ ..............................................................................
10)
«‘∏’∑”
sin sin 3sin 5 = 0 (sin 5sin )sin 3 = 0 2 sin 3 cos 2sin 3 = 0 sin 3 (2 cos 21) = 0 sin 3 = 0
, , , , , , 0 , 60 , 120 , 180 , 240 , 300 , 360 ................................................................................
0 180 360 540 720 900 1080 3 = ................................................................................
=
0 2 cos 21 = ................................................................................
1 cos 2 = ................................................................................ 2 120 240 480 600 2 = ................................................................................
,
,
,
60 , 120 , 240 , 300 = ................................................................................ 0 , 60 , 120 , 180 , 240 , 300 , 360 ¥—ßπ—Èπ ‡∑à“°—∫ .................................................................................. 11)
«‘∏’∑”
1 tan 1 tan o tan 45 tan o 1 tan 45 tan
=
3
=
3
tan (45 ) =
3
45 =
=
60 , 240 .............................. 15 , 195 ..............................
(tan 45 = 1) tan (AB) =
tan A tan B 1 tan A tan B
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
230 12)
tan 3 = tan
«‘∏’∑”
sin 3 cos 3
sin cos
=
sin 3 cos = cos 3 sin 0 sin 3 cos cos 3 sin = .................................................. 0 sin (3) = .................................................. 0 sin 2 = ..................................................
, , , , 0 , 90 , 180 , 270 , 360 ..................................................
0 180 360 540 720 2 = ..................................................
= 13)
sin x = 2 sin(60 x)
«‘∏’∑”
sin x = 2(sin 60 cos xcos 60 sin x) sin x = sin x =
°”Àπ¥ tan 40 54 = 0.8660
Ê 3 ˆ 1 2Á cos x sin x˜ 2 Ë 2 ¯ 3 cos xsin x
3 cos x 2 sin x = .................................................. sin x cos x
3 = .................................................. 2
3 tan x = .................................................. 2 = 0.8660 ..................................................
,
54 220 54 x = 40 .................................................. 14)
«‘∏’∑”
sin (60 ) = 2 cos (30 ) sin cos 60 cos sin 60 = 2(cos cos 30 sin sin 30 )
3 1 sin cos = 2 2
Ê 3 ˆ 1 2Á cos sin ˜ 2 Ë 2 ¯
3 1 sin cos = 2 2 3 sin = 2 sin = cos
3 cos ........................................ 2
tan =
1 ........................................ 3
=
30 , 210 ........................................
3 cos sin
3 2 ........................................ 2 3
231
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 sin (20 ) = cos (20 )
15)
«‘∏’∑”
sin cos 20 cos sin 20 = cos cos 20 sin sin 20 o o o o sin cos 20 cos sin 20 cos cos 20 sin sin 20 = o o cos cos 20 cos cos 20 o o sin sin 20 sin sin 20 = 1 o cos cos 20o cos cos 20 tan tan 20 = 1........................... tan tan 20 ........................................ tan tan tan 20 = ................................. 1tan 20 ........................................ tan (1tan 20 ) = ................................. 1tan 20 ........................................ 0 1tan 20 = ................................. tan (1tan 20 )(....................) 0 (tan 1)(1tan 20 ) = ................................. 0 tan 1 = ..........
0 1tan 20 = ..........
À√◊Õ
1 tan 20 = ..........
1 tan = .................... 45 , 225 = ....................
‡π◊ËÕß®“° tan 20 0
45 ·≈– 225 ¥—ßπ—Èπ ‡∑à“°—∫ .................................................. 16)
2 sin x = tan x 0 x 360
«‘∏’∑”
2 sin x = tan x
,
2 sin x =
sin x cos x π 0 cos x
,
2 sin x cos x = sin x 2 sin x cos xsin x = sin x(2 cos x1) sin x = x =
0 .......... 0 , 180 , 360 ..............................
0 .................... 0 = .................... À√◊Õ 2 cos x1
=
cos x = x =
0 , 60 , 180 , 300 , 360 ¥—ßπ—Èπ x ‡∑à“°—∫ .................................................. 17)
2 cos x = cot x 0 x 360
«‘∏’∑”
2 cos x = cot x
,
2 cos x =
cos x sin x π 0 sin x
2 sin x cos x = cos x 2 sin x cos xcos x = 0 2 sin x1 = 0 cos x(..................)
,
0 .......... 1 ..........
2
60 , 300 ....................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
232 0 cos x = ....................
0 2 sin x1 = .................... 1 sin x = .................... 2
À√◊Õ
90 270 x = ....................
,
30 150 x = ....................
,
30 , 90 , 150 , 270 ¥—ßπ—Èπ x ‡∑à“°—∫ .................................................. 18)
cos (3x75 ) =
3 0 x 360 2
«‘∏’∑”
cos (3x75 ) =
3 2
,
3x75 = 30 330 390 690 750
,
,
,
,
,
,
,
,
405 465 765 825 3x = 105 ..................................................
,
,
,
,
135 155 255 275 x = 35 .................................................. 19)
sin xcos x = 0
«‘∏’∑”
sin xcos x = 0 sin x = cos x
sin x cos x
=
cos x , cos x π 0 cos x
tan x = 1
,
135 315 x = .................... 2
20)
2 tan = 3 sec
«‘∏’∑”
2 tan = 3 sec
2
2
2(sec 1) = 3 sec 2
2 sec 23 sec = 0 2
2 sec 3 sec 2 = 0 2 sec 1 sec 2 (....................)(....................) 0 0 2 sec 1 = ............
0 sec 2 = .................
À√◊Õ
1 sec = ............ 2
2 sec = ................. 60 , 300 = .................
‡π◊ËÕß®“° sec 1 À√◊Õ sec 1 60 , 360 ¥—ßπ—Èπ = ................. 2
21)
4 sin = 3
«‘∏’∑”
4 sin = 3
2 2
sin =
3 4
233
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
3 sin = ............... 2 p 2p = ............... 3 3
3 sin = ............... 2
3 sin = ............... 2 4 p 5p = ............... 3 3
À√◊Õ
,
,
§à“∑—Ë«‰ª¢Õß ∑’Ë∑”„Àâ ¡°“√‡ªìπ®√‘ߧ◊Õ p 2p 5p 4p 2np , 2np.........., 2np.........., 2np.......... ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ 3 3 3 3
p
np §”µÕ∫√«¡§◊Õ .................... ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ 3 2
22)
tan 1 = 0
«‘∏’∑”
tan 1 = 0
2
2
tan = 1 tan = 1
....................
tan =
=
1 .................... p 5p , .................... 4 4
À√◊Õ
tan =
=
1 .................... 3p 7p .................... , 4 4
§à“∑—Ë«‰ª¢Õß ∑’Ë∑”„Àâ ¡°“√‡ªìπ®√‘ߧ◊Õ p 5p 3p 7p .........., 2np.........., 2np.........., 2np.......... ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ 4 4 4 4 p np §”µÕ∫√«¡§◊Õ .................... ‡¡◊ Õ Ë n ‡ªìπ®”π«π‡µÁ¡ 4
2np
23)
tan sin tan = 0
«‘∏’∑”
tan sin tan = 0 sin 1 tan (....................) = 0 0 tan = ...............
À√◊Õ
0, p = ...............
0 sin 1 = ............... 1 sin = ............... 3p = ............... 2
´÷Ëß tan À“§à“‰¡à ‰¥â 2np0 , ............... 2npp ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ §à“∑—Ë«‰ª¢Õß ∑’Ë∑”„Àâ ¡°“√‡ªìπ®√‘ß §◊Õ ............... np ¥—ßπ—Èπ = .......... ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ 24)
cos 2 = cos
«‘∏’∑”
cos 2 = cos 2
2 cos 1 = cos
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
234 2
2 cos cos 1 = 0 (2 cos 1)(cos 1) = 0 ..................................................................................................................................................... 0 2 cos 1 = ............... 1 cos = ............... 2 2p 4p = ............... 3 3
0 cos 1 = ...............
À√◊Õ
1 cos = ............... 0, 2p = ...............
,
4p 2p p §à“∑—Ë«‰ª¢Õß §◊Õ 2np.........., 2np.........., 2np, 2np.......... ‡¡◊ËÕ n ‡ªìπ®”π«π‡µÁ¡ 3 3
2.10 °Æ¢Õß‚§‰´πå·≈–‰´πå °Æ¢Õß‚§‰´πå „π√Ÿª “¡‡À≈’ˬ¡ ABC „¥Ê ∂â“ a, b, c ‡ªì𧫓¡¬“«¢Õߥâ“πµ√ߢⓡ¡ÿ¡ A, B ·≈– C µ“¡≈”¥—∫ 2
2
2
2
2
2
2
2
2
a = b c 2bc cos A b = c a 2ca cos B c = a b 2ab cos C Ÿ
µ—«Õ¬à“ß∑’Ë 1 °”Àπ¥ A = 60 , b = 40 ·≈– c = 60 ®ßÀ“ a 2 2 2 «‘∏’∑” ®“°°Æ¢Õß‚§‰´πå a = b c 2bc cos A 2
2
= 40 60 2(40)(60) cos 60 Ê 1ˆ = 1 6003 6002(2 400) Á ˜ Ë 2¯
,
,
,
= 2 800
,
¥—ßπ—Èπ
a =
20 7
= 52.915
µÕ∫
235
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 µ—«Õ¬à“ß∑’Ë 2 °”Àπ¥ a = 4, b = «‘∏’∑” ®“°°Æ¢Õß‚§‰´πå
2 19 b
2
·≈– c = 6 ®ßÀ“ 2
Ÿ
B
2
= c a 2ca cos B
cos B = =
2
2
2
2
c a b 2ca
2
6 4 (2 19) 2(6)( 4)
2
36 16 76 48 1 = 2 =
Ÿ
B = 120
¥—ßπ—Èπ
µÕ∫
°“√„™â‡§√◊ËÕߧ”π«≥À“§«“¡¬“«¢Õߥâ“π·≈–¢π“¥¢Õß¡ÿ¡ ®“°
a
2
2
2
= b c 2bc cos A
À“ a ®“°‡§√◊ËÕߧ”π«≥¥—ßπ’È b x
2
c x
2
2 b c A cos = 2
cos A =
2
b c a 2 bc
x
2
À“¢π“¥¢Õß A ®“°‡§√◊ËÕߧ”π«≥¥—ßπ’È ( b x
2
∂â“
c x
2
a x
a b = sin A sin B
2
)
1
2 b c = cos
®–‰¥â a =
b sin A sin B
À“ a ®“°‡§√◊ËÕߧ”π«≥¥—ßπ’È b A sin
B sin
=
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
236
a b = sin A sin B
∂â“
®–‰¥â sin A =
a sin B b
À“¢π“¥¢Õß A ®“°‡§√◊ËÕߧ”π«≥¥—ßπ’È a B sin b =
æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC
1
sin
1 bc sin A 2 1 (20)(17.93) sin 45 2
= =
= (10)(17.93)(0.707) = 126.7651
µ“√“ßÀπ૬
°“√„™â‡§√◊ËÕߧ”π«≥ ®“°µ—«Õ¬à“ß∑’Ë 1 2
a
2
2
= b c 2b cos A 2
2
= 40 60 2(40)(60) cos 60 a =
2
2
40 60 2( 40)(60) cos 60
o
≈”¥—∫°“√„™â‡§√◊ËÕߧ”π«≥À“ a ¥—ßπ’È 2
a = 40 x
60 x
2
2 40 60 60.0 cos =
= 52.91502622
®“°µ—«Õ¬à“ß∑’Ë 2 2
cos A =
2
6 4 (2 19) 2(6)( 4)
cos A = 6 x
2
4 x
2
2
2 x
2
19
2 6 4 = = 0.5 1
A = 0.5 = 2nd cos = 120
®“°µ—«Õ¬à“ß∑’Ë 3 c =
20 sin 60 o sin 75
o
c = 20 60 sin 75 sin = = 17.93150944
x
2
x
=
x
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
°‘®°√√¡∑’Ë 2.10.1 1.
®ß„™â°Æ¢Õß‚§‰´πåÀ“§«“¡¬“«¢Õߥâ“π∑’ˇÀ≈◊Õ ‚¥¬„™â‡§√◊ËÕߧ”π«≥ 1) a = 45, b = 67, C = 35 2
= a b 2ab cos C
2
= 45 67 2(45)(67) cos 35
c
c
2
2
c = 2)
a =
2
2
= 20 40 2(20)(40) cos 28
b =
2
2
=
a = b = 2
=
2
=
a a
a = c = 2
=
2
=
b = 6)
c = b b
24.23394007 ........................................................................................ 10.5, c = 40.8, A = 120
= b c 2bc cos A
a
b
2
2
a
b
2
2
c =
5)
39.68013575 ........................................................................................ 20, b = 40, C = 28
= a b 2ab cos C
c
4)
2
2
c
3)
2
2
=
2
=
b =
2
2
2
(10.5) (40.8) 2(10.5)(40.8)cos 120 ........................................................................................ 46.93921601 ........................................................................................ 12.9, c = 15.3, A = 104.2 2 2 b c 2bc cos A ........................................................................................ 2 2 (12.9 )(15.3) 2(12.9)(15.3) cos 104.2 ........................................................................................ 22.30095598 ........................................................................................ 38, a = 42, B = 135.3 2 2 c a 2ca cos B ........................................................................................ 2 2 38 42 2(38)(42)cos 135.3 ........................................................................................ 74.00589112 ........................................................................................ 3.49, a = 3.54, B = 5.4 2 2 c a 2ca cos B ........................................................................................ 2 2 (3.49) (3.54) 2(3.49)(3.54) cos 5.4 ........................................................................................ 0.334903425 ........................................................................................
237
238
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
2.
®ß„™â°Æ¢Õß‚§‰´πåÀ“¢π“¥¢Õß¡ÿ¡∑’Ë „À≠à∑’Ë ÿ¥ (µ√ߢⓡ¥â“π∑’ˬ“«∑’Ë ÿ¥) ‚¥¬„™â‡§√◊ËÕߧ”π«≥ 1) a = 7.23, b = 6.00, c = 8.61 2) a = 16.0, b = 17.0, c = 18.0 2
®“°
2
a b c 2ab
cos C =
2
2
2
®“°
2
(7.23) 6 (8.61) 2(7.23)(6)
=
cos C =
2
2
= 0.163 80.6 .................................
C =
3.
2
0.40625 ................................. 66 .................................
,
2
b c a ................................. 2bc
= =
2
2
2
cos B =
c a b ................................. 2ca
14 10 18 .................................
=
500 300 600 .................................
0.1 ................................. 95.7 .................................
= 0.067
2
2
®“°
2
2(14)(10)
B =
2
2
2
2(500)(300)
................................. 93.8 .................................
®ßÀ“ à«π∑’ˇÀ≈◊Õ¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ‚¥¬„™â°Æ¢Õß‚§‰´πå ‡¡◊ËÕ°”Àπ¥ 1) a = 4.21, b = 1.84, C = 30.7 2 2 2 ®“° c = a b 2ab cos C 2
2
= (4.21) (1.84) 2(4.21)(1.84) cos 30.7 =
7.788180727 ....................................................... 2.79 .......................................................
c =
2
2
b c a 2 bc
cos A =
2
2
2
(1.84) (2.79) ( 4.21) 2(1.84)(2.79)
=
2
= 0.638382422
....................................................... 129.7 ....................................................... 180 129.7 30.7 ....................................................... 19.6 .......................................................
A = B = =
2) a = 2 b = 3 c = 4
,
,
2
®“°
cos A =
2
=
,
cos A =
A =
2
16 17 18
................................. 2(16)(17)
4) a = 300 b = 600 c = 500
,
2
®“°
2
=
C =
3) a = 18 b = 14 c = 10
,
2
a b c 2ab
2
b c a 2 bc
2
2
cos B =
2
c a b 2ca
2
239
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 2
= = A = C = =
2
3 4 2 2(3)( 4)
2
2
=
2
4 2 3 2( 4)(2)
2
= 0.6875
24.833 ........................................ 29 ........................................ 180 29 46.6 ........................................ 104.4 ........................................
B =
°Æ¢Õ߉´πå „π√Ÿª “¡‡À≈’ˬ¡ ABC „¥Ê ∂â“ a, b, A, B ·≈– C µ“¡≈”¥—∫
c
........................................ 46.6 ........................................
‡ªì𧫓¡¬“«¢Õߥâ“πµ√ߢⓡ¡ÿ¡
sin A sin B sin C = = a b c
Y
Y
C(b cos A, b sin A)
C(b cos A, b sin A) b
a
A(0, 0)
c
a
B(c, 0)
X
B
æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC
= =
Ÿ
µ—«Õ¬à“ß∑’Ë 3 °”Àπ¥ A «‘∏’∑” ®“°°Æ¢Õ߉´πå ‡π◊ËÕß®“°
Ÿ
= 45 C = 60
,
sin B b Ÿ
Ÿ
Ÿ
A B C Ÿ
=
X
·≈– b = 20 ®ßÀ“ c ·≈–æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC sin C c
= 180
Ÿ
B = 75 o o sin 75 sin 60 = 20 c c =
A
1 b sin Ac 2 1 bc sin A 2
45 B 60 = 180
·∑π§à“®–‰¥â
c
b
20 sin 60 o sin 75
o
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
240
=
3 2 31 2 2
20
=
10 3 2 2 31 31 31
=
10 6 ( 3 1) ( 6 = 2.449
,
3 = 1.732)
= 17.93
µÕ∫
°‘®°√√¡∑’Ë 2.10.2 1.
®ßÀ“ à«π∑’ˇÀ≈◊Õ¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ‚¥¬„™â°Æ¢Õ߉´πå·≈–‡§√◊ËÕߧ”π«≥ 100 1) A = 32 , B = 48 , a = 10 ®–‰¥â C = .................... ®“° sina A = sinb B ®“° sina A = sinc C b = =
a sin B sin A
c =
a sin C .............................. sin A
=
10 sin 100 .............................. o
o
10 sin 48 .............................. o sin 32
c =
b sin C .............................. sin B
=
40 sin 110 .............................. o
..............................
a sin A
=
40 sin 20 .............................. o
=
17.9 ..............................
= 49.1
b sin A sin B o 15 sin 50 = .............................. o sin 40 = 17.9
..............................
b sin B b sin A .............................. sin B
sin 50
a =
=
a =
o
.............................. 90 3) A = 50 , B = 40 , b = 15 ®–‰¥â C = .................... ®“° sina A = sinb B ®“°
sin 32
= 18.6
= 14
.............................. 20 2) C = 110 , B = 50 , b = 40 ®–‰¥â A = .................... ®“° ®“° sinb B = sinc C
o
b sin B
o
sin 50
c sin C b sin C c = sin B o 15 sin 90 = .............................. o sin 40 =
= 23.3
..............................
241
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 4) a = 4 b = 8 A =
,
,
30 ....................
®“°
a sin A
®–‰¥â
sin B =
b sin B b sin A a
sin B =
8 sin 30 .............................. 4
=
5) C = 70 b = 100 c = 100
,
=
a = =
c =
c sin C a sin C sin A
c =
4 sin 60 .............................. o
=
o
sin 30 6.9 =
.............................. 90 .............................. 180 30 90 = 60 .............................................
C =
a sin A
®–‰¥â
1
B =
®“°
a sin A
o
=
,
®“°
..............................
70 40 ®–‰¥â B = ...................., A = ....................
b sin B b sin A .............................. sin B
100 sin 40 o sin 70
o
..............................
= 68.4
..............................
2.11 °“√À“√–¬–∑“ß·≈–§«“¡ Ÿß ¡ÿ¡°â¡ ¡ÿ¡‡ß¬
°“√«—¥¡ÿ¡ N
¡ÿ¡‡ß¬
A
∑‘»∑“ß Õß»“ ∑‘»µ–«—πµ°‡©’¬ß‡Àπ◊Õ √–¥—∫ “¬µ“
¡ÿ¡°â¡
A
E
∑‘»∑“ß Õß»“ ∑‘»µ–«—πÕÕ°‡©’¬ß„µâ
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
242
°‘®°√√¡∑’Ë 2.11 1.
„Àâπ—°‡√’¬πÀ“§à“¢Õß x ´÷Ëß·∑𧫓¡¬“«¢Õߥâ“π 1)
2) 30
a
100
¡.
a
32 x
100
58
¡.
¡.
¡.
55
= sin 58 = a =
58 32 = 26 ..............................
=
100 a
=
100 .............................. o sin 58
=
x = = =
a
sin 32 a sin o sin 32
.............................................
= 97.56
‡¡µ√ .............................................
=
a =
o
o 100 sin 26 o o sin 58 sin 32 100 0.4384 0.8480 0.5299
55 30 = 25 .............................. 90 30 = 120 ..............................
‚¥¬°Æ¢Õ߉´πå 100 sin
‚¥¬°Æ¢Õ߉´πå x sin
x
a sin
100 sin sin
x = a sin 55
·∑π§à“ a,
x =
o 100 sin sin 55 sin o o 100 sin 120 sin 55 o sin 25
=
.............................................
=
1000.86600.8192 ............................................. 0.4226
= 167.87
‡¡µ√ .............................................
2.
‡§√◊ËÕß∫‘π 2 ≈” ∫‘πÕÕ°®“° π“¡∫‘π‡¥’¬«°—π·≈–‡«≈“‡¥’¬«°—π ≈”Àπ÷Ëß∫‘π‰ª∑“ß∑‘»µ–«—π ÕÕ°‡©’¬ß‡Àπ◊ե⫬§«“¡‡√Á« 400 ‰¡≈åµàÕ™—Ë«‚¡ß ≈”∑’Ë Õß∫‘π‰ª∑“ß∑‘»µ–«—πµ°¥â«¬§«“¡‡√Á« 300 ‰¡≈åµàÕ™—Ë«‚¡ß ®ßÀ“«à“À≈—ß®“°∑’ˇ§√◊ËÕß∫‘π Õß≈”π’ÈÕÕ°®“° π“¡∫‘π‰ª‰¥â 2 ™—Ë«‚¡ß ®– Õ¬ŸàÀà“ß°—π°’Ë ‰¡≈å
243
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 «‘∏’∑”
B
800
‰¡≈å
135 C 600
‰¡≈å A
„Àâ A ‡ªìπµ”·ÀπàߢÕß π“¡∫‘π ‡¡◊ËÕ‡«≈“ºà“π‰ª 2 ™—Ë«‚¡ß 800 ‡§√◊ËÕß∫‘π≈”·√°Õ¬Ÿà∑’Ë B ·≈– AB = ...............‰¡≈å 600 ‡§√◊ËÕß∫‘π≈”∑’Ë ÕßÕ¬Ÿà∑’Ë C ·≈– AC = ...............‰¡≈å Ÿ 135 BAC = ............... ®“°°Æ¢Õß‚§‰´πå 2
BC
2
2
= AB AC 2ABAC cos A 2
2
= 800 600 2(800)(600) cos 135 = 1678822.51 .............................. BC = 1295.693833 .............................. 1,295.69 ¥—ßπ—Èπ ‡¡◊ËÕ‡«≈“ºà“π‰ª 2 ™—Ë«‚¡ß ‡§√◊ËÕß∫‘πÕ¬ŸàÀà“ß°—π ..................... °‘‚≈‡¡µ√ 3.
∑’Ë®ÿ¥®ÿ¥Àπ÷Ëß¡Õ߇ÀÁπ¬Õ¥‡ “‰øøÑ“·√ß Ÿß‡ªìπ¡ÿ¡‡ß¬ 30 ‡¡◊ËÕ‡¥‘πµ√߇¢â“‰ª¬—߇ “‰øøÑ“·√ß Ÿßπ—Èπ‡ªìπ√–¬–∑“ß 40 ‡¡µ√ ®–¡Õ߇ÀÁπ¬Õ¥‡ “‰øøÑ“·√ß Ÿß‡ªìπ¡ÿ¡‡ß¬ 75 ®ßÀ“«à“‡ “ ‰øøÑ“·√ß Ÿßµâππ—Èπ Ÿß‡∑à“‰√ «‘∏’∑” „Àâ A ‡ªìπ®ÿ¥∑’Ë¡Õ߬ե‡ “‰øøÑ“·√ß Ÿß§√—Èß·√°‡ªìπ¡ÿ¡‡ß¬ 30 B ‡ªìπ®ÿ¥∑’Ë¡Õ߬ե‡ “‰øøÑ“·√ß Ÿß§√—Èß∑’Ë Õ߇ªìπ¡ÿ¡‡ß¬ 75 CD ‡ªì𧫓¡ Ÿß¢Õ߇ “‰øøÑ“·√ß Ÿß D
75 A
30
40
¡.
AB =
„π√Ÿª “¡‡À≈’ˬ¡ ACD; tan 30
=
B
C
40 ............... ‡¡µ√ CD AC
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
244
1 = 3
CD AB BC
1 = 3
CD 40 BC
„π√Ÿª “¡‡À≈’ˬ¡ BCD; tan 75
=
2 3
=
CD BC CD BC
CD .............................. 2
BC =
3
1 = 3
·∑π§à“ BC „π (1);
CD CD 40 2 3
CD 2 3
=
3 CD
CD 2 3 2 3 2 3
=
3 CD
40(2 3 )CD =
3 CD
40 40
.....(1)
40 .............. 40 .............. CD ..............
= ( 3 2 3 )CD
CD ..............
= 10( 3 1)
............................................ 2( 3 1)CD ............................................ 31 20 ............................................
= =
31
31
............................................ 102.732 = ............................................ 27.32 = ............................................ 27.32 ‡¡µ√ ¥—ßπ—Èπ ‡ “‰øøÑ“·√ß Ÿß Ÿß ............... E
Ÿ
Ÿ
BAC = DAE
®“°√Ÿª
4.
Ÿ
36
BAD
= =
BC = 9 CD = 72 DE = 36
D
,
,
®ßÀ“ AB 72
«‘∏’∑”
A
C 9 B
tan ( ) =
BE =
BE AB
=
............... ·≈–„Àâ AB = x 117 117 ............... x
245
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 BD AB BC tan = AB
tan =
=
81 ............... x
=
9 ............... x
tan ( ) =
¥—ßπ—Èπ
117 x
117 x
tan tan 1 tan tan
=
81 9 x x 81 9 1 x x
=
90 x 2 x 729 x
117 x
2
=
2
117(x 729) = 2
117x 117729 = 2
=
2
=
27x x
= x = ⬇
¥—ßπ—Èπ
AB =
90 x x x 2 729 2
90x .............................. 2 90x .............................. 117729 .............................. 11727 .............................. 8139 .............................. 9 39 .............................. 56.20 .............................. 56.2 ..............................
¡.
™“¬§πÀπ÷Ë߬◊πÕ¬Ÿà∫πÀπ⓺“´÷Ëß Ÿß 300 ‡¡µ√ —߇°µ‡ÀÁπ‡√◊Õ Õß≈”§◊Õ A ·≈– B ‡√◊Õ A Õ¬Ÿà ∑‘»µ–«—πµ°¢Õ߇¢“ ‡ªìπ¡ÿ¡°¥≈ß 28 ‡√◊Õ B Õ¬Ÿà „π∑‘»∑“ß 20 ∑‘»µ–«—πµ°‡©’¬ß„µâ ‡ªìπ¡ÿ¡ °¥≈ß 19 ®ßÀ“√–¬–∑“ß√–À«à“߇√◊Õ∑—Èß Õß «‘∏’∑” „Àâ P ‡ªìπ®ÿ¥ —߇°µ‡√◊Õ A ·≈– B P PC ‡ªì𧫓¡ Ÿß¢ÕßÀπ⓺“ N 300 ‡¡µ√ PC = .......... 300
5.
2
tan 28 =
28
A
C 19
20
AC =
PC AC PC
o tan 28
=
300 ........................................ o tan 28
=
300 = 564.23 ........................................ 0.5317
B
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
246
N P P
A
28
B
C
„π∑”πÕ߇¥’¬«°—π BC
= =
Ÿ
ACB
A
19
300 tan 19
C
B
o
300 = 871.33 ............................................. 0.3443
= 90 20 = 70
.......................
‚¥¬°Æ¢Õß‚§‰´πå AB
2
2
2
= AC BC 2ACBC cos 70 2
2
= (564.23) (871.33) 2564.23871.330.3420
6.
.......................................................................................... ,296.1821 = 741 .......................................................................................... AB = 860.98 .......................................................................................... 861 ¥—ßπ—Èπ ‡√◊Õ∑—Èß ÕßÀà“ß°—πª√–¡“≥ ............... ‡¡µ√ ®“°®ÿ¥ Õß®ÿ¥§◊Õ A ·≈– B Õ¬Ÿà∫πÕ“§“√‡¥’¬«°—πÀà“ß°—π 19 ‡¡µ√ „π·π«µ—Èß ¡Õ߇ÀÁπ √∂¬πµå§—πÀπ÷Ë߮ելŸàÀπâ“Õ“§“√™—Èπ≈à“߇ªìπ¡ÿ¡°â¡ 58 ·≈– 34 µ“¡≈”¥—∫ ®ßÀ“«à“®ÿ¥ B Õ¬Ÿà Ÿß®“°æ◊Èπ√–¥—∫‡¥’¬«°—∫√∂¬πµå‡∑à“‰√ «‘∏’∑” ®“°√Ÿª C ‡ªìπµ”·ÀπàߢÕß√∂¬πµå A 58 19 AB = .......... ‡¡µ√ 19 ¡. „π√Ÿª “¡‡À≈’ˬ¡ ABC 34
Ÿ
B
BAC Ÿ
ACB C
‚¥¬°Æ¢Õ߉´πå
90 58 = 32 = .............................. 58 34 = 24 = ..............................
D
BC sin 32
o
=
AB o sin 24
o 19 sin 32 BC = .................................... o sin 24
„π√Ÿª “¡‡À≈’ˬ¡ BCD;
BD = BC sin 34 o o 19 sin 32 sin 34 = .................................... o sin 24
247
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 =
190.52990.5592 .................................................. 0.4067
= 13.84
.................................................. 13.8..... ‡¡µ√ ¥—ßπ—Èπ ®ÿ¥ B Õ¬Ÿà Ÿß®“°æ◊Èπ .......... 7.
∂ππ¢÷Èπ‡π‘π∑”¡ÿ¡°—∫·π«√“∫ 32 ™“¬§πÀπ÷Ëß∂’∫®—°√¬“π®“°®ÿ¥∑’Ë¡Õ߇ÀÁπ¬Õ¥‡¢“·ÀàßÀπ÷Ëß ‡ªìπ¡ÿ¡‡ß¬ 47 ‡¡◊ËÕ∂’∫®—°√¬“π¢÷Èπ‡π‘π‰ª‰¥â 1 °‘‚≈‡¡µ√ ¡Õ߇ÀÁπ¬Õ¥‡¢“‡ªìπ¡ÿ¡‡ß¬ 77 ®ßÀ“«à“¬Õ¥‡¢“Õ¬Ÿà Ÿß®“°·π«√“∫‡∑à“‰√ °”Àπ¥„Àâ sin 47 = 0.731 «‘∏’∑”
„Àâ P ‡ªìπ¬Õ¥‡¢“ A ·≈– B ‡ªìπ®ÿ¥ —߇°µ¬Õ¥‡¢“ Ÿ Ÿ PAC = 47 , BAC = 32
P
77 B
E
Ÿ
PDC
=
Ÿ
P BE = 77
AB = 1 000
,
47 32 A
77 D
‡¡µ√
C
„Àâ x ‡ªì𧫓¡ Ÿß¢Õ߬ե‡¢“®“°æ◊Èπ√“∫ Ÿ sin PAC „π√Ÿª “¡‡À≈’ˬ¡ PAC; x = PA ........................................ sin 47 = PA ........................................
„π√Ÿª “¡‡À≈’ˬ¡ PAB;
Ÿ
PAB Ÿ
APB
¥—ßπ—Èπ ®“°°Æ¢Õ߉´πå
Ÿ
ABP PA Ÿ
sin ABP
= 15 = 47 32 ........................................ = 77 47 ........................................ = 30 = 135 ........................................ AB = Ÿ sin APB Ÿ
AB sin A BP PA = ........................................ Ÿ sin A P B o 1000 sin 135 = ........................................ o sin 30
Ê 1ˆ 1000Á ˜ Ë 2¯ = ........................................ 1 2 1000 2 = ........................................
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
248
x = = =
PA sin 47 ........................................ 1000 2 0.731 ........................................ 731 2 ........................................
731 2 ¥—ßπ—Èπ ¬Õ¥‡¢“ Ÿß ....................‡¡µ√ 8.
¡ÿ¡¬°¢÷Èπ¢Õ߬եµ÷°À≈—ßÀπ÷Ëß®“°®ÿ¥ A ·≈– B ∫πæ◊Èπ√“∫‡ªìπ 22 ·≈– 40.5 µ“¡≈”¥—∫ ∂â“ A ·≈– B Àà“ß°—π 180 ‡¡µ√ ®ßÀ“§«“¡ Ÿß¢Õ߬եµ÷°À≈—ßπ’È «‘∏’∑” ®“°√Ÿª A ·≈– B ‡ªìπ®ÿ¥ —߇°µ¢Õ߬եµ÷° D CD ‡ªì𧫓¡ Ÿß¢Õ߬եµ÷° Ÿ
= 22
CAD 22
A
40.5
Ÿ
CBD = 40.5
C
B
AB = 180 Ÿ
ADB = BD o = sin 22
„π√Ÿª “¡‡À≈’ˬ¡ ABD, ‚¥¬°Æ¢Õ߉´πå
BD =
‡¡µ√
18.5 ..............................
AB sin 18.5
o
.......... o 180 sin 22 .............................................
„π√Ÿª “¡‡À≈’ˬ¡ BCD;
o sin 18.5 CD = BD sin 40.5
·∑π§à“ BD ®–‰¥â
CD =
o
o
180 sin 22 sin 40.5 ............................................. o sin 18.5
=
1800.37460.6494 ............................................. 0.3173
= 138.00
.............................................
¥—ßπ—Èπ ¬Õ¥µ÷° Ÿß ............... ‡¡µ√ 138 ·≈– B Õ¬Ÿà∫πæ◊Èπ√“∫Àà“ß°—π 200 ‡¡µ√ ∑’Ë®ÿ¥ B ¡Õ߇ÀÁπ®ÿ¥ P ∫π¬Õ¥µ÷°‡ªìπ¡ÿ¡¬°¢÷Èπ 25 Ÿ Ÿ ∂â“ PQ ‡ªì𧫓¡ Ÿß¢Õßµ÷° BAQ = 60 ·≈– ABQ = 40 ®ßÀ“ 1) §«“¡ Ÿß¢Õßµ÷° 2) ¡ÿ¡¬°¢÷Èπ¢Õ߬եµ÷°∑’Ë®ÿ¥ P P «‘∏’∑”
9. A
P
A
P
Q Q
25
Q
A
25
60
40
B
®“°√Ÿª
Ÿ
AQB = 180 60 40 = 80
............................................
B
249
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1 1)
„π√Ÿª “¡‡À≈’ˬ¡ ABQ ‚¥¬°Æ¢Õ߉´πå BQ sin 60
„π√Ÿª “¡‡À≈’ˬ¡ BPQ;
o
=
200 sin 80
o o
BQ =
200 sin 60 .................................................. o
=
200 sin 60 tan 25 .................................................. o
=
2000.86600.4663 ..................................................
sin 80 PQ = BQ tan 25
o
o
sin 80
0.9848
= 82.01
..................................................
2)
82 ¥—ßπ—Èπ µ÷° Ÿß ............... ‡¡µ√ „Àâ ‡ªìπ¡ÿ¡¬°¢÷Èπ¢Õ߬եµ÷°∑’Ë®ÿ¥ A tan =
PQ AQ
„π√Ÿª “¡‡À≈’ˬ¡ ABQ ‚¥¬°Æ¢Õ߉´πå AQ sin 40
o
=
200 sin 80
o
o 200 sin 40 AQ = ............................................................ o sin 80 o o o 200 sin 60 tan 25 sin 80 tan = ............................................................ o o 200 sin 40 sin 80 0.6283 = ............................................................ 32.1 = ............................................................ 32.1 ¥—ßπ—Èπ ¡ÿ¡¬°¢÷Èπ∑’Ë®ÿ¥ A ‡∑à“°—∫ ...............
10.
™“¬§πÀπ÷ßË Õ¬Ÿà „π‡√◊Õ —߇°µ‡ÀÁπ¥«ß‰ø∫π¬Õ¥Õ“§“√√‘¡Ωíßò ·¡àπÈ”‰ª∑“ß∑‘»‡Àπ◊Õ‡ªìπ¡ÿ¡¬°¢÷πÈ ‡¡◊ËÕ‡√◊Õ·≈àπ‰ª∑“ß∑‘»µ–«—πÕÕ° 500 ‡¡µ√ —߇°µ‡ÀÁπ¥«ß‰ø∫π¬Õ¥Õ“§“√‡¥‘¡‡ªìπ¡ÿ¡¬° ¢÷Èπ ®ßÀ“ 1) §«“¡ Ÿß¢Õߥ«ß‰ø¬Õ¥Õ“§“√ 2) ¢π“¥¢Õß¡ÿ¡∫πÕ“§“√√‘¡Ωíòß·¡àπÈ”√Õß√—∫√–¬–∑“ß∑’ˇ√◊Õ‡≈àπ «‘∏’∑”
1)
„À⥫߉ø¬Õ¥Õ“§“√Õ¬Ÿà Ÿß h ‡¡µ√ „π√Ÿª “¡‡À≈’ˬ¡ ACD;
·∫∫Ωñ° §≥‘µ»“ µ√å‡æ‘Ë¡‡µ‘¡ ¡.5 ¿“§‡√’¬π∑’Ë 1
250
D
h AC
N
= tan
AC = C
„π√Ÿª “¡‡À≈’ˬ¡ BCD;
h BC B
A 500
h .................... tan
= tan
BC =
¡.
h .................... tan
„π√Ÿª “¡‡À≈’ˬ¡ ABC; 2
2
h ˆ Ê Ë tan ¯
2
BC AC
Ê h ˆ Á ˜ Ë tan ¯
2
2
2
=
500 ....................
1 1 ˆ 2 Ê h Á 2 2 ˜ Ë tan tan ¯
=
2 500 ....................
2 2 2 Ê tan tan ˆ h Á 2 2 ˜ Ë tan tan ¯
=
500 ....................
........................................ ........................................
2)
= AB
2
h =
500 tan tan .................................................. 2 2
BC =
h ..................................................
Ÿ
sin ACB
tan tan
tan
=
500 ..................................................
=
500 ..................................................
=
500 tan .................................................. h
BC
h tan
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