Cercul Trigonometric (1)

Principalele valori ale funct¸iilor sin x ¸si cos x

Funct¸iile trigonometrice ale multiplilor unui unghi

y

√ ! 1 3 − , 2 2 ! √ √ 2 2 − , 2 2 2π ! √ 3 3 1 3π , − 2 2 4 120◦ 5π ◦ 135 6 150◦ (−1, 0)

(0, 1)

√ ! 1 3 , 2 2

π 2 90◦

π 3 60◦

ctg2α =

√ √ ! 2 2 , 2 2 √

3 1 , 2 2

π 4

!

cos 3α = cos α(4 cos2 α − 3) tg3α =

3tgα−tg 3 α 1−3tg 2 α

ctg3α =

ctg 3 α−3ctgα 3ctg 2 α−1

Pentru funct¸iile tg α2 ¸si ctg a2 se folosesc uneori ¸si formulele

◦

30

(1, 0)

360 0◦ ◦

ctg 2 α−1 2ctgα

sin 3α = sin α(3 − 4 sin2 α)

Funct¸iile trigonometrice ale jum˘ at˘ a¸tii unui unghi q q 1−cos α α α tg = ± sin α2 = ± 1−cos 2 q 2 q1+cos α 1+cos α α ctg α2 = ± 1−cos cos α2 = ± 1+cos α 2

π 6

45◦

180◦

π

sin 2α = 2 sin α cos α cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2sin2 α 2 tg α tg2α = 1−tg 2α

2π

x

tg α2 = tg α2 =

sin α 1+cos α 1−cos α sin α

ctg α2 = ctg α2 =

1+cos α sin α sin α 1−cos α

De asemenea sunt utile ¸si formulele 210◦

−

√

3 1 ,− 2 2

!

7π 6 5π 4

√ ! 2 2 − ,− 2 2 √

225◦ 240◦ 4π 3

330◦ 11π 6

315◦ 300 270◦

◦

5π 3

3π 2

√ ! 1 3 − ,− 2 2

7π 4

√

3 1 ,− 2 2

√ ! 2 2 ,− 2 2 √ ! 1 3 ,− 2 2 √

(0, −1)

Formule fundamentale sin2 α + cos2 α = 1;

tg α =

sin α cos α

Funct¸iile trigonometrice ale sumei ¸si diferent¸ei de unghiuri sin(α + β) = sin α · cos β + cos α · sin β sin(α − β) = sin α · cos β − cos α · sin β cos(α + β) = cos α · cos β − sin α · sin β cos(α − β) = cos α · cos β + sin α · sin β

tgα+tgβ 1−tgα·tgβ tg α−tg β tg(α − β) = 1−tg α·tg β α·ctg β−1 ctg(α + β) = ctg ctg α+ctg β α·ctg β+1 ctg(α − β) = ctg ctg β−ctg α

tg(α + β) =

!

sin α = 2 sin α2 cos α2 ; cos α = cos2 α2 − sin2 α2 ;

1 − cosα = 2 sin2 α2 ; 1 + cos α = 2 cos2 α2

Exprimarea funct¸iilor trigonometrice ale unghiului α cu ajutorul tg α2 (formulele universale) sin α = cos α =

2tg α 2 1+tg 2 α 2 1−tg 2 α 2 2 1+tg α 2

2tg α 2 1−tg 2 α 2 1−tg 2 α = 2tg α 2 2

tg α = ctg a

Formule pentru transformarea unor sume ¸si diferent¸e de funct¸ii trigonometrice ˆın produs. α−β sin α + sin β = 2 sin α+β 2 · cos 2 α+β sin α − sin β = 2 sin α−β 2 · cos 2 α+β cos α + cos β = 2 cos 2 · cos α−β 2 β−α cos α − cos β = 2 sin α+β · sin 2 2

tg α ± tg β =

sin(α±β) cos α·cos β

sin(α+β) sin α·sin β sin(β−α) ctg α − ctg β = sin α·sin β cos(α+β) tg α − ctg β = − cos α cos β 1 − cos α = 2sin2 α2 1 + cos β = 2cos2 α2

ctg α + ctg β =

Formule pentru transformarea unor produse de funct¸ii trigonometrice ˆın sume: tg β+tg β sin α · cos β = 12 [sin(α + β) + sin(α − β)] tg α · tg β = ctg α+ctg β cosα · cosβ = 12 [cos(α + β) + cos(α − β)]

sinα · sinβ = 12 [cos(α − β) − cos(α + β)]

ctg α+ctg β tg α+tg β α+tg β = ctg tg α+ctg β

ctg α · ctg β = ctg α · tg β