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Philippine Philippine Handbook in Chemical Engineering
Trigonometry Trigonometry and and Solid Mensuration Mensuration Azucena Puertollano
A2. TRIGONOMETRY TRIGONO METRY AND SOLID SOLI D MENSURATION A2.1. TRIGONOMETRY A2.1.1 Angles 1. Angle - set of points determined by two rays or finite line segments, L 1 (initial side) and L2 (terminal side) having a common end point O. 2. Standard position of an Angle - obtained by taking its vertex at the origin of the rectanglar coordinate system and letting the initial side L1 coincide with the positive x!axis. 3. Positive Angle - formed by a conterclockwise rotation of L1 to its terminal position L2. 4. egative Angle - formed by a clockwise rotation of L1 to L2. 5. Straight Angle - sides lie on the same straight line bt extend in opposite direction from its vertex, e.g., 1"## . 6. Co!terminal Co!terminal Angles - any two angles having the same initial and terminal sides, no matter the amont or direction of rotation of L 1 before coming to position L2 in a specified $adrant, e.g., %2## and !&###. 7. "uadrantal Angle - terminal side lies on a coordinate axis, e.g ., '## or 1"##. 8. #ne degree $% & ' - the measre of the central angle of a circle sbtended by (or opposite to) an arc e$al in length to the radis of the circle. 9. Conversion (actors - radians 1"##* +# mintes deg * +# seconds min. 1. Measures of Angle ) cte ngle # # - - '# # Obtse ngle '# # - - 1"# # 0 '# # omplementary ngles / 0 1"# # pplementary ngles / ight ngle '# # &# #
%5#
+# #
'# #
Special Angles
&##
2
&
+#
#
1 3 &#o!+#o!'#o 4
2 %5
%5#
#
1
+ % & 2
1 3 %5o!%5o!'#o4
7sally, no nits n its are sed for radian measre, i.e., 5 means 5 radians, not NOTE6 7sally, 5 degrees.
2 ! 1
Philippine Philippine Handbook in Chemical Engineering
Trigonometry Trigonometry and and Solid Mensuration Mensuration Azucena Puertollano
11. *ength 1. *ength of a circular arc s on a circle of radis r sbtending a central angle of radian measre 6 s = r
(2 ! 1) 12. Area 12. Area A of a circular circular sector #
s
A =
r r
1 2
r 2 ,
, in radians
(2 ! 2)
irclar sector 13. Angular 13. Angular speed of a wheel rotating at a constant rate of n revoltions per minte !the angle generated per nit time by a line segment from the center of the wheel to a point on its circmference6 n (2 ! &) pointt on the the circ circm mfe fere rence nce of a whee wheell of radis radis r 6 distance 14. *inear 14. *inear speed of a poin r traveled by the point per nit time, . NOTE6 8hereas the linear speed depends on the diameter of the wheel, the latter is irrelevant in finding the anglar speed . A2.1.2 T$%g&n&'e($%! )*n!(%&ns
ngle T+"le A2-1. 9rigonometric :nctions of an cte ngle )UN,TION
DE)INITION
)ORMULA
ine
O;; ? ?
tan
osecant
? O;;
cot
Hypotenuse !
"
Opposite side
9he ight 9riangle 2 ! 2
b c a c b a c
b c a a b
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+, #ther Trigonometric (unctions,
vers 1 cos cov ers 1 sin ex sec sec 1 hav
1 2
vers
(2 ! %) (2 ! 5) (2 ! +) (2 ! @)
NOTE# 1. 9he vales of the six trigonometric fnctions are positive for every acte angle since the lengths of the sides of a right triangle are positive real nmbers. 2. 9he hypotense is always greater than the adAacent or opposite side. &. 9he following fnctions are reciprocal of each other6 sine and cosecant, cosine and secant, tangent and cotangent. %. ccording to the ;ythagorean 9heorem6 2 2 2 Hyp adj. opp (2 ! ") 5. 9he adAacent and opposite sides are perpendiclar to each other* hence, their inclded angle is '# # . -, Trigonometric (unction of the .uadrantal angle of coordinates of point ; on a nit circle6
(#,1)
# sin1"#
#
# cos 1"#
1
# tan1"#
#
in the standard position in terms
y 1 r # cos '# x # r # y tan '# x
sin '##
y
(1,#) (!1,#)
r 01
x
sin #
#
#
#
1
#
#
cos # tan #
# sin 2@#
1
# cos 2@#
#
# tan 2@#
(#,!1)
9he Badrantal ngles
2 ! &
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/, To determine the value of the function of any angle 0 1. Cet the vale of the fnction of the reference angle R , i.e., the acte angle that the terminal side of makes with the x!axis.
2. ;refix the sign of the fnction of the angle in the specified $adrant HH
HH
H sin
sin cos
cos
tan
tan
sin
sin
cos
cos
tan
tan
H
y
θ R
θ R
x
θ x
y
y
x
θ R
θ R
HHH
HJ
HHH
y
x
HJ
lgebraic igns of 9rigonometric :nctions 9he eference ngles, D # # &. :or # and &+# , obtain its coterminal angle # # &+# # * then perform steps (1) and (2), sing the reference angle R and the coterminal angle, respectively. EFG;LE6 :ind the exact vale of OL79HOI6
cos 2#
+
cos 2#
+
cos( +## # ) cos( &+# # 2%# # )
cos +# #
oterminal angle of the given Een +n O T$%g&n&'e($%! )*n!(%&ns
a. Even fnction f ( x ) f ( x) 6 cos( x) cos( x) sec( x) sec( x) b. Odd fnction f ( x) f ( x) 6 sin( x ) sin( x) tan( x) tan( x) csc( x) csc( x) cot( x) cot( x )
2 ! %
R
1
2
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NOTE6 9he graphs of even fnctions are symmetrical with respect to the y!axis* the odd fnctions, symmetrical with respect to the origin.
2 ! 5
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Trigonometry and Solid Mensuration Azucena Puertollano
A2.1.3. T$%g&n&'e($%! )&$'*l+s %, Trigonometric Substitution x a sin for
EFG;LE6
a x 2
2
2
a a sin 2
x a tan for
2
2
x a sec for #
2
2
and a K #
a cos a cos 2
2
and a K #
& or and a K # 2 2
+, The (undamental 1dentities
1. Reciprocal Identities sin
1
(2 1#)
csc 1 cos sec 1 tan cot
(2 11) (2 12)
2. an!ent " #otan!ent Identities tan cot
sin cos cos
(2 1&) (2 1%)
sin
$. %ytha!orean Identities sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 csc 2
(2 15) (2 1+) (2 1@)
-, Addition and Subtraction (ormulas $u and v are real numbers' sin u v sin u cos v cos u sin v cos u v cos u cos v sin u sin v
tan u v
tan u tan v
(2 1") (2 1') (2 2#)
1 tan u tan v
2 ! +
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/, 2eduction of trigonometric functions of sm or difference of angles into fnctions of alone. Al(e$n+(%e I# 7se additionsbtraction formlas Al(e$n+(%e II# :ind the terminal side of the given smdifference of angles. 9hen apply the steps for determining the vale of the fnction of any angle previosly otlined. 5 in terms of a trigonometric fnction EFG;LE6 Express cos 2 of alone.
OL79HOI6 ALTERNATI/E I
cos
5
5
5
cos # sin 1 sin cos cos sin sin 2 2 2
ALTERNATI/E II 0See )%g*$e
5 2
5 cos cos R sin 2 2
3, Cofunction (ormulas
sin 2 sin cos 2 tan cot 2 cos
(2 21) (2 2&) (2 25)
csc 2 csc sec 2 cot tan 2
sec
(2 22) (2 2%) (2 2+)
4, 5ouble!Angle (ormulas sin 2u 2 sin u cos u cos 2u cos 2 u sin 2 u 1 2 sin 2 u 2 cos 2 u 1 2 tan u tan 2u 1 tan 2 u
2 ! @
(2 2@) (2 2") (2 2') (2 &#) (2 &1)
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Trigonometry and Solid Mensuration Azucena Puertollano
6, Half!Angle 1dentities (so!called becase the nmber is one!half the angle 2 on the right side of the e$ality sign) sin 2 u
cos 2 u
tan 2 u
1 cos 2u
(2 &2)
2 1 cos 2u
(2 &&)
2 1 cos 2u
(2 &%)
1 cos 2u
7, Half!Angle (ormulas sin cos tan
v
2 v
(2 &5)
2
2 v
1 cos v 1 cos v
(2 &+)
2
2
1 cos v
1 cos v
1 cos v
sin v
sin v
(2 &@)
1 cos v
NOTE# hoose the algebraic sign depending on the $adrant containing the angle
9hs if
v
2
.
v v is in $adrant HHH, cos is negative (!). 2 2
8, Product!to!Sum (ormulas sin u cos v
12 sin u v sin u v
(2 &")
cos u sin v
12 sin u v sin u v
(2 &')
cos u cos v
12 cos u v cos u v
sin u sin v
12 cos u v cos u v
(2 %#) (2 %1)
%&, Sum!to!Product (ormulas sin sin 2 sin
sin sin 2 cos
cos
2
cos cos 2 cos
sin
2
2
2
cos 2 2 cos cos 2 sin sin 2 2
A2.1.4. Ine$se T$%g&n&'e($%! )*n!(%&ns %, 19E2SE S1E $or arcsine' (:CT1# # en&(e " sin 1 &$ +$!s%n
2 ! "
(2 %2) (2 %&) (2 %%) (2 %5)
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DE)INITION#
y arcsin x if
and only if x sin y ,
DOMAIN# 1 x 1 * GENERAL SOLUTION O) sin y = x6 y ; n< = $!%'n arcsin >? n @1 @2 @3 @,, y ; arcsin >? with range of
RIN,IAL /ALUE#
2
arcsin x
vales.
2
+, 19E2SE C#S1E $or arccosine' (:CT1# # en&(e " cos DE)INITION#
1
&$ +$!!&s
y arccos x if and only if cos y x ,
1 x 1 and range6 # arccos x
DOMAIN#
GENERAL SOLUTION of cos y x # y
2n
?
arccos > ? n
1 ?
2 ?
3 ?
,,,,,
RIN,IAL /ALUE# y arccos x -, 19E2SE TABET $or arctangent' (:CT1# # en&(e " tan 1 &$ +$!(+n DE)INITION#
y arctan x if and only if tan y x ,
x DOMAIN# GENERAL SOLUTION of tan y x # y
n
arctan > ? n
?
1 ?
2 ?
RIN,IAL /ALUE# y arctan x , with a range6
3 ?
,,,,,
2
arctan x
2
/, 19E2SE C#TABET $or arccotangent' (:CT1# # en&(e " cot +$!!&( DE)INITION#
y arc cot x if and only if x cot y , x DOMAIN# GENERAL SOLUTION of cot y x # y
n
arc cot > ? n
2 ! '
?
1 ?
2 ?
3 ?
,,,,,
1
&$
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Trigonometry and Solid Mensuration Azucena Puertollano
RIN,IAL /ALUE# y arc cot x , with a range6 # arc cot x 3, 19E2SE C#SECAT $or arccosecant' (:CT1# # en&(e " csc +$!!s! DE)INITION#
1
&$
y csc 1 x if and only if csc y x ,
1 DOMAIN# GENERAL SOLUTION O) csc y x # x
y
1
n
RIN,IAL /ALUE# y
n
arc csc > ? n
?
1 ?
2 ?
3 ?
arc csc x , with a range6 2
arc csc x
2
4, 19E2SE SECAT $or arcsecant' (:CT1# # en&(e " sec DE)INITION#
,,,,,
1
&$ +$!se!
y sec 1 x if and only if sec y x ,
x 1 DOMAIN# GENERAL SOLUTION O) sec y x # y
2n
RIN,IAL /ALUE# y
arc sec > ? n
?
1 ?
2 ?
3 ?
,,,,,
arc sec x , with a range6 # y
NOTE6 M Arcsin xN means Man an!le &hose sine is xN and Marccosine xN., Man an!le &hose 1 & , arccos , arctan & , etc, refer to the same angle cosine is xN, and so on. 9hs, arcsin 2 2 # y0+# .
A2.1.5. S&l*(%&ns & T$%+ngles %, Solving the 2ight Triangle 1. An!le of 'levation angle that the line of sight (to an elevated obAect) makes with the horiontal line, sally at eye level of observer. 2. An!le of (epression similarly defined, except that the obAect sighted is below the horiontal line or eye level. NOTE# 9he sm of the interior angles of any triangle6
1"# # Civen6 one side (a) b or c) and any acte angle , or any 2 sides 2 ! 1#
(2 %+)
Philippine Handbook in Chemical Engineering
Trigonometry and Solid Mensuration Azucena Puertollano a
(c b)(c b) c sin b tan
b
(c a )(c a #) c cos a tan
(2 %@) (2 %")
+, Solution of #bli.ue Triangles O"l%*e T$%+ngle does not contain a right angle. olving obli$e triangles mean finding the measres of the angles , and corresponding to the vertices , P and , and measres of the sides opposite them designated by a, b and c, respectively. -, *a of Sine a
sin
b
sin
c
(2 %')
sin
,+se I# Civen the measres of two angles and a side, where
1"# # ,+se II# (ambigos) Civen the measre of 2 sides a and b and an acte angle opposite a &ss%"%l%( 10See )%g*$e# a b sin
y
C a
SOLUTION# Io triangle formed becase P does not intersect the x!axis to complete a triangle &ss%"%l%( 20See )%g*$e# a b sin
b
D
y
C
SOLUTION# One right triangle formed
b
a
A
&ss%"%l%( 30See )%g*$e# b sin a b SOLUTION# 2 possible triangles formed, P1 and P2, since side a intersect the x!axis at P1 and P2
2 ! 11
b sin x
D
y
C b
a
a
b s%n
A
&ss%"%l%( 4# a b SOLUTION# Only one possible triangle* side a intersects the x!axis at P only. Hf a b , an isosceles triangle is obtained.
x
b sin
A
D+
D%
x
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,+se III# One angle ( ) is obtse6 the measre of the side a opposite the angle is greater than the measres of the other sides. &ss%"%l%( 1# a b SOLUTION# One triangle is formed
e G$+> & (>e T>$ee R&&(s & Un%(
NOTE# 9he three roots are e$ally spaced on a circle of radis
&
1 1.
A2.2. SOLID MENSURATION A2.2.1. A$e+s & l+ne )%g*$es 1. S.uare#
d e 2 *
A e 2. 2ectangle#
d
2
d is length of diagonal and e is the edge
d 2
b2
1 2
h 2 , A bh b d 2 b 2 h d 2 h 2
3. Parallelogram $opposite sides parallel # A bh , h is the perpendiclar distance between parallel sides 1 A 2 s1 ( s1 b)( s1 h)( s1 d 1 ) , where s1 2 (b h d 1 ) A 2 s2 ( s2 b)( s2 h)( s2 d 2 ) ,
where s 2
1
2 (b h d 2 )
d 1 is short diagonal, d 2 is long diagonal
4. 2hombus $e.uilateral parallelogram'# A
1 2
ab , a and b are length of the diagonals.
2 ! 15
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5. Triangle#
Trigonometry and Solid Mensuration Azucena Puertollano
A 12 bh ,
h is length of line dropped perpendiclarly from one vertex to the line where its opposite side lies.
A
s ( s
a )( s
b)( s
c)
where s
,
1
2 a bc
and
a) b
and c are the three sides of the triangle 1 6. Trapezoid $four sides? to parallel' # A 2 (b1
b2 )h , h is the perpendiclar distance s
between shorter and longer bases, b1 and b2, respectively. 7. Circle#
2 # 2 ? A r d % %
# 2 r d
2
r
# = circmference ) r 0 radis, d 0 diameter 8. Sector of a Circle# A 12 rs 12 r , s 0 arc length s 0 r , in radians 2
b h
r
9. Segment of a Circle# A rs bh 1 2
,%$!*l+$ Se!(&$
1 2
s
,%$!*l+$ Seg'en( 1. Ellipse# (See )%g*$e
A ab
b
length of maAor axis 0 2 a length of minor axis 0 2 b
# (approximate) 0 2
a
2
b2 2
a
2 Parabolic Segment # 0See )%g*$e A & bh
2 Length of arc ('0 %h
2 b 2h %h 2 2 2 b b 1 2 ln b 2 2 2h 2
E
h
5
b 2 ! 1+
(
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Trigonometry and Solid Mensuration Azucena Puertollano
11. 2egular Polygon of n sides# A R
1 2
1 %
1"## , l 0 length of each side n
nl cot 2
1"## , R 0 radis of circmscribed circle n
csc
1"## r cot n , r 0 radis of the inscribed circle 2 1
&+# # n
l 2 r tan
2 R sin
2 (n 2)1"# #
2
n
12. Area of polygon inscribed in a circle of radius 2 $See (igure' #
&+# # sin A 2 n nR 2
l
inscribed circle
r 2
13. Perimeter of inscribed polygon$See (igure'0
% 2 nR sin
1"# # n
14. Area of polygon circumscribed about a circle of radius r $See (igure'# 2 A nr tan
1"# # n
inscribed circle 2 ! 1@
l
r 2
circmscribed circle
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A2.2.2. S*$+!e A$e+s +n /&l*'e & S&l%s 1. Polyhedron solid bonded by polygons* if reglar, all faces are reglar polygons $ $ $ $
Eges & + &l>e$&n intersections of the bonding planes +!es portions of the bonding planes enclosed by the edges. e$(%!es intersections of the edges. (e($+>e$&n for faces are e$ilateral triangles6 A f (area of one face)
& %
e 2 , where
Jolme of 9etrahedron, 1
2 12
e
e 0 length of an edge
&
adis of inscribed sphere in a tetrahedron, r 0 e
$
>e:+>e$&n cbe, radis of inscribed sphere , r
$
&!(+>e$&n eight faces that are e$ilateral triangles A f
1
$
$
& 2 e % 2 &
e
A 2 & e
&
2
2
radis of inscribed sphere 0
Gene$+l )&$'*l+s &$ &l>e$&ns A n f A f 1 1& n f A f r ,*"e6 9otal rea, A
+ e 12
+e 2 ,
Jolme, 1 e &
+ e +
where n f = nuber of faces
e 0 edge length
& & d , d 0 length of diagonal of cbe '
$
Re!(+ng*l+$ +$+llele%e# A 2(ab bc ac) 1 abc , where a) b and c are length, width and height, respectively
$
$%s' ! polyhedron with two e$al and parallel base polygons and lateral faces which are parallelograms6 Lateral rea, A 3 e% r where e 0 lateral edge % r 0 perimeter of right section 1 *h Ar e where * 0 area of the base Jolme, h 0 altitde Ar 0 area of the right section
2. Cylinders
2 ! 1"
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Trigonometry and Solid Mensuration Azucena Puertollano
R%g>( ,%$!*l+$ ,l%ne$ 6 A 3 2 rh , * r 2 , 1 r 2 h 2 2 @&ll&= ,l%ne$s 6 1 h( R r ) , where R and r are external and internal radii T$*n!+(e R%g>( ,%$!*l+$ ,l%ne$ 0See )%g*$e# A 3 2 rh , 1 r 2 h , where h 12 (h1 h2 ) h1
h h2 h
h1
h2
3. Pyramid
A 3 0 sm of areas of trianglar faces 1 1& *h NOTE# :or a reglar pright pyramid (vertex directly above the center of the bases),
12 % b 34 where % b 0 perimeter of the base* 34 0 slant height or altitde of one
A 3 face
4. 2ight Circular Cone
A 3
r34 12 #34
1 1& r h 2
where # 0 circmference of the base
34 0 slant height of cone
5. (rustum $ :rstm of eglar ;yramid (obtained by ctting off the portion containing the vertex with a plane, sally parallel to the base)6
A 3
% 1 % 2 2
1
34
A A2 1 &
A1
A2 h
% 1 0 perimeter of bigger base % 2 0 perimeter of smaller base 34 0 slant height or altitde of one trapeoidal face A1 0 area of bigger base A2 0 area of smaller base $
:rstm of a ight irclar one6 # # 2 A 3 1 34 R r 34 2 1
1 &
A A 1
2
A1
A2 h
where R and r are radii of the bigger base and smaller base, respectively.r 1 h1
6. Sphere $See (igure' srface area, A % R 2
h
h2 2
2 ! 1'
5
r 2
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volme, 1 %& R
Trigonometry and Solid Mensuration Azucena Puertollano
1+ ( & 2 2 (spherical sector) 2& R h 1+ ( h &
(spherical segment with one base)
h1
(spherical segment with two bases)
+ h +
&r
h12
&r
&r 22 h22
2 2
2 1
&ne ! portion of the srface of a sphere inclded between two parallel planes A( /one) 2 Rh (h
7. Ellipsoid every section perpendiclar to the axis of the solid is an ellipse
1 %& abc , where a) b, and c are the length of the semi!axes. 8. Torus obtained by rotating a circle of radis r abot a line whose distance is R5r from the center of the circle. 1 2 2 Rr 2
srface area 0 % 2 Rr
9. Spheroids $
$&l+(e S>e$&% obtained by rotating an ellipse abot its maAor axis6
srface area 2 b 2 ab 2
e
sin
1
e , where e is the eccentricity (e-1).
1 %& ab 2 $
O"l+(e S>e$&% by rotating ellipse abot its minor axis6
b 2 1 e 2 ln a 2 srface area , where e is the eccentricity (e-1). e 1 e 2 1 %& a b
A2.3. S@ERI,AL TRIGONOMETRY A2.3.1. T>e S>e$%!+l T$%+ngle 1. Circle intersection of a plane with a sphere 2. Breat Circle formed when the plane passes throgh the center of the sphere &. Small Circle formed when the plane passes throgh any point other than the center.
2 ! 2#
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%. A>is of Circle line throgh the center of the sphere perpendiclar to the plane of the great circle and pierces the sphere at two extreme points called the poles. 5. Spherical Triangle ADC (see %g*$e) bonded by three arcs (P, P and ) of great circles* composed of three angles (, P and ) and their opposite sides (a, b and c, respectively). P
a
O
c
b
+. Trihedral Angle Mspace cornerN opposite to the spherical triangle* vertex is at the center O of the sphere. 9he plane angles , and are the face angles of the trihedral angle. @. Sides of the Spherical Triangle measred by the corresponding face angles of the trihedral angle6 a is measred by *O# or * b by AO# or * and c by AO* or . ". Angles of the Spherical Triangle measred by the corresponding dihedral angles of the trihedral angle. :or example, angle A is measred by the dihedral angle whose edge is OA and bonded by the faces AO* and AO# . '. #bli.ue spherical triangle 6 does not have a right angle ('#S) NOTE6 ince the face angles , and are the central angles of the respective sides a) b) and c of the spherical triangle, a , b and c in anglar measres. 9herefore, the sides of the spherical triangle have trigonometric fnctions.
*imitations on the sides of a spherical triangle0 1. # o - a 7 b 7 c - &+# o 2. a) b) or c shold not be greater than 1"# o
*imitations on the angles of a spherical triangle 1. Io angle ( A) *) or # ) of a spherical triangle is e$al to or greater than 1"# o 2. 1"# o - A 7 * 7 # - 5%#o Theorems on spherical triangles
2 ! 21
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H. HH.
Trigonometry and Solid Mensuration Azucena Puertollano
9he sm of any two sides of a spherical triangle is greater than the third side 9he largest angle is opposite to the longest side* the smallest angle is opposite to the shortest side.
A2.3.2. Gene$+l L+=s The Cosine *a for Sides 6 9he cosine of any side of a spherical triangle e$als the prodct of the cosines of the other two sides pls the prodct of the sines of these two sides mltiplied by the cosines of their inclded angle. cos a cos b cos c sin b sin c cos A cos b cos c cos a sin c sin a cos * cos c cos a cos b sin a sin b cos #
The Cosine *a for Angles6 9he cosine of any angle of a spherical triangle is e$al to the prodct of the sines of the other two angles mltiplied by the cosine of their inclded side mins the prodct of the cosine of the other two angles. cos A sin * sin # # cos a cos * cos # cos * sin # sin A # cos b cos # cos A cos # sin A sin * # cos c cos A cos *
The Sine *a 6 Hn a spherical triangle, the sines of the angles are proportional to the sines of the opposite sides. sin A sin a
sin * sin b
sin # sin c
Haversine *a for Angles0
Let s = T ,a 7 b 7 c
sin( s b) sin( s c ) sin b sin c sin( s c) sin( s a) hav* sin c sin a sin( s a) sin( s b) hav# sin a sin b havA
Haversine *a for Sides0 hav a hav (b c) sin b sin c havA hav b hav (c a) sin c sin a hav* hav c hav (a b) sin a sin b hav #
A b
A2.3.3. S&l*(%&ns & S>e$%!+l T$%+ngles
c O
2ight Spherical Triangle
2 ! 22
# *
a
Philippine Handbook in Chemical Engineering
Trigonometry and Solid Mensuration Azucena Puertollano
Hf angles * and # 0see %g*$e are both '##!angles, then the faces AO* and AO# are perpendiclar to plane *O# , 9he edge of intersection OA of planes AO* and AO# are likewise perpendiclar to the planes *O# * ths, angles AO* and AO# are right angles and arcs A* and A# are $adrants of great circles, i.e., '##!arcs. right spherical triangle contains one '##!angle* birectanglar triangle, two right angles* and trirectanglar triangle, three right angles. .uadrantal triangle has one side e$al to a $adrant or '##* bi$adrantal, two sides each e$al to '##* tri$adrantal, three '##!sides. NOTE# 1. Hn a birectanglar triangle like A*# , the sides (b and c) opposite the right angles * and # are $adrants ('##). 2. 9he third angle A has the same measre as its opposite side a. &. Hf each of the three angles is '##, each side is also '## or $adrantal. Hn this case, the triangle is its own polar. %. right spherical triangle (only one angle e$al '##) has its opposite side different from '##.
A
N+%e$Bs R*les 9he :ive nglar Bantities of the ight pherical 9riangle (Left :igre) rranged in a icle (ight :igre)
A b
c
c b * a
'#
#
#
*
9he bars over the letters A) * and c are read Mthe complement ofN, example6 A means M'## AN. #ircular parts anglar $antities a) b) c A , and * ( 0 '## is exclded in the circle). 8iddle part any given part Adjacent parts parts contigos with or adAacent to any given part Opposite parts two non!adAacent or non!contigos parts ,
Example6 Hn the right figre, if and * and a are opposite to it.
A
is the middle or given part,
2 ! 2&
c
and b are adAacent to
A
,
Philippine Handbook in Chemical Engineering
Trigonometry and Solid Mensuration Azucena Puertollano
apiers 2ules State0
H.
9he sine of any middle part is e$al to the prodct of the cosines of the opposite parts. 9he sine of any middle part is e$al to the prodct of the tangents of the adAacent parts.
HH.
NOTE6 Msine middle 0 cos opposite 0 tan adAacentN E:+'le# Obtain the formla for tan b sing IapierQs rles.
S&l*(%&n#
le HH is directly applicable, since it involves tangents of adAacent parts. :or b to be adAacent, choose either 9 or a as middle part (conslt the circle). Hf a is chosen, b and * are adAacent to it* then from le HH6 sin a 0 tan b tan * olving for tan b6 tan b
sin a
tan *
sin a tan('# o
* )
sin a cot *
sin a tan *
2 ! 2%
Philippine Handbook in Chemical Engineering
Trigonometry and Solid Mensuration Azucena Puertollano
2eferences0 E>8>, .avid E. ;enney. 1''%. Calculus ith Analytic Beometry, ;rentice!e GoivreQs 9heorem, 2!1& ellipsoid, 2!1' even fnction, 2!% frstm, 2!1' fndamental identities, 2!5
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