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2 210>7
Fig. 9.4
9.18 Show that (a) sin u (b) sin A (c) tan 4x (d) cos 6
1 (e) sin2 2 u
2 sin 1 u cos 2
1 cos 2A 2 A sin 8x 1 cos 8x 1 2 sin2 3
1 2 (1 1 cos u), cos 2 2 u 1 2 (1
cos u)
1 2 u.
(a) This is obtained from sin 2 (b) This is obtained from 1 u 2 (c) This is obtained from 2 u (d) This is obtained from cos 2
2 sin 1 A 1
by putting a
cos u by putting 2 8x.
sin u by putting cos u 1 2 sin2 by putting
3 . 1 cos u and cos 1 u 2 2 1 A cos u . 2
(e) These formulas are obtained by squaring sin 1 u 2
CHAPTER 9 Trigonometric Functions of Two Angles
9.19 Express (a) sin 3a in terms of sin a and (b) cos 4a in terms of cos a.
(a) sin 3a sin (2a 2 sin a (1 (b) cos 4a cos 2(2a) a) sin 2a cos a (1 (1 1 sin a)
cos 2a sin a 2 sin2 a) sin a 2 sin a cos2 a 3 sin a 1)
(2 sin a cos a) cos a 2 cos 2a
2 sin2 a) sin a 8 cos2 a 1
2 sin a) sin a
4 sin a 8 cos4 a
2(2 cos a
9.20 Prove cos 2x
cos4 x
cos4 x
sin4 x.
sin4 x (cos2 x sin2 x)(cos2 x sin2 x) cos2 x sin2 x cos 2x
Prove 1
1 2 sin 2x
sin 3x sin x
cos3 x cos x
cos 3x . cos x
( sin x 1 cos x)( sin2 x sin x 1 sin x cos x cos x
1 2 (2 sin x cos x)
sin3 x sin x
cos2 x) 1
1 2 sin 2x
sin x cos x
9.22 Prove cos u
sin (u 30 )
sin (u
cos (u
30 )
60 )
cos (u
60 ).
cos u sin 30 ) 1 cos u 2 1 cos u 2 (cos u cos 60 23 sin u 2 sin u sin 60 ) cos u
(sin u cos 30 23 sin u 2
9.23 Prove cos x
tan2 1 x 2 tan2 1 x 2
sin2 2 x cos2 1 x 2
1 sec2 2 x 1
a1 sin2 2 x cos2 1 x 2
1 tan2 2 x 1 tan2 2 x
b cos2 1 x 2 cos2 1 x 2
1 sin2 2 x
1 sec2 2 x cos2 1 x 2
cos x
9.24 Prove 2 tan 2x
cos x cos x sin x sin x
cos x cos x cos x cos x
sin x sin x sin x sin x
cos x cos x
sin x sin x
sin x)2 ( cos x sin x)2 ( cos x ( cos x sin x)( cos x sin x) sin 2x) ( cos 2x ( cos 2x 2 sin x cos x cos 2x sin 2x 4 sin x cos x cos 2x sin 2x 2 sin 2x cos 2x 2 tan 2x
2 sin x cos x
sin 2x)
9.25 Prove sin 4 A
1 2 cos 2
1 8 cos 4A.
sin 4 A
( sin 2 A)2
1 4 a1
cos 2A 2 R 2 1
2 cos 2A 4
3 8 1 2 cos 2
cos2 2A A
1 8 cos 4A
2 cos 2A
cos 4A R 2
9.26 Prove tan6 x
tan6 x
tan4 x sec2 x
tan4 x tan2 x tan x sec x tan4 x sec2 x
tan2 x sec2 x
tan4 x (sec2 x tan x (sec x tan2 x sec2 x
sec2 x
1) 1)
tan2 x tan2 x tan2 x sec2 x tan2 x
tan4 x sec2 x tan x sec x 1
sec2 x
CHAPTER 9 Trigonometric Functions of Two Angles
9.27 When A
Since C
sin 2A
180
180 , show that sin 2A
(A B), sin 2C sin 2A sin 2A sin 2A
sin 2B
sin 2B sin 2B sin 2B
sin 2C
2(A B)
4 sin A sin B sin C.
B)] cos 2A sin 2B cos 2A)
sin 2B
sin [360 sin 2(A
sin 2A cos 2B (sin 2B)(1 2 sin 2B sin2 A
(sin 2A)(1
cos 2B)
2 sin 2A sin2 B
4 sin A cos A sin2 B 4 sin A sin B sin (A 4 sin A sin B sin [180
4 sin B cos B sin2 A cos A sin B) B)] 4 sin A sin B sin C B) (A
4 sin A sin B (sin A cos B
9.28 When A
Since C tan A
180 tan B tan A tan A ( tan A
180 , show that tan A
(A B), tan B tan B tan B)a tan [180 (A tan C
tan B
tan C
tan A tan B tan C.
tan A ( tan A
tan B
tan (A 1
B) 1 b tan A tan B
tan A tan B 1 tan A tan B 1 tan A tan B b tan A tan B B)
tan B) a1 1
tan A tan B (A
tan A tan B tan A tan B B)] tan A tan B tan C
tan A tan B tan (A
tan A tan B tan [180
SUPPLEMENTARY PROBLEMS
9.29 Find the values of the sine, cosine, and tangent of (a) 75 and (b) 255 . Ans. (a) !6 4 !2 !6 , !2 4 ,2 !3 (b) !6 4 !2 , !6 4 !2 ,2 !3
Find the values of the sine, cosine, and tangent of (a) 7 /12 and (b) 11 /12. Ans. (a) !6 4 !2 !6 , !2 4 , 2 !3 (b) !6 4 !2 , !6 4 !2 , 2 !3
Rewrite each expression as a single function of an angle. (a) sin 173 cos 82 cos 173 sin 82 (b) cos 86 cos 73 sin 86 sin 73 tan 87 tan 21 (c) 1 tan 87 tan 21 (d) sin 87 cos 87 (e) 2 cos2 151 1 (f) 1 2 sin2 100 tan 42 (g) 1 tan2 42 2 (h) cos 81 sin2 81 sin 56 (i) 1 cos 56 1 cos 76 (j) 2 A Ans. sin 255 Ans. cos 13 Ans. tan 66 Ans.
sin 174
Ans. cos 302 Ans. cos 200 Ans.
tan 84
Ans. cos 162 Ans. tan 28 Ans. cos 38
CHAPTER 9 Trigonometric Functions of Two Angles
Find the values of sin ( (a) sin (b) sin (c) cos (d) sin 3/5, cos 8/17, tan 12/13, cot 1/3, sin 2/5,
), cos ( 5/13, 5/12, and and
), and tan ( in quadrant I in quadrant I in quadrant II,
), given: Ans. 63/65, 16/65, 63/16
Ans. 171/221, 140/221, 171/140 in quadrant III Ans. 36/325, 323/325, 36/323
24/7,
in quadrant I, Ans. 4 22 15
in quadrant II 221 , 2 221 2 242 4 22 , 15 2 2 242 ), given: Ans. 16/65, 63/65, 16/63 Ans. 21/221, 220/221, 21/220 Ans. 204/325, 253/325, 204/253 !21 4 !2 2 !42 2 25 !2 82 9 !21 25 22 9 221 82
Find the values of sin ( (a) (b) (c) (d) sin sin cos sin
), cos (
), and tan (
3/5, sin 5/13, and in quadrant I 8/17, tan 5/12, and in quadrant I 12/13, cot 24/7, in quadrant II, is a quadrant I 1/3, sin 2/5, in quadrant II, in quadrant I Ans. 4!2 15 !21 , 2 !42 15 2 ,
Prove: (a) sin ( (b) cos ( (c) tan (45 (d) tan (a cot (a ) ) u) b) b) ) y) y) u) sin ( cos ( 1 1 ) ) tan u tan u 2 cos 2 cos sin cos
tan2 b tan2 a 1 tan2 a tan2 b tan [( ) ] tan a tan b 1 tan a tan b tan g tan a tan b tan g tan b tan g tan g tan a
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