CHAPTER 9
590
ANALYTIC TRIGONOMETRY
13-18 • Use an appropriate half-angle formula to find the
exact value of the expression.
13. sin 15°
14. tan 15°
15. cos 22.5°
16. tan^
17. sin—
10
41. 2 sin 52.5° sin 97.5°
42. 3 cos 37.5° cos 7.5°
43. cos 37.5° sin 7.5°
44. sin 75° + sin 15°
TT
45. cos 255° -cos 195°
18. cos —-
1-9
41-46 • Find the value of the product or sum.
COS
~\2
\2
^ "
47-64 • Prove the identity.
47. cos2 5* — sin2 5x = cos 10*
19-24 • Simplify the expression by using a double-angle formula or a half-angle formula.
48. sin 8* = 2 sin 4x cos 4x
19. (a) 2 sin 18° cos 18°
49. (sin x + cos x)2 = 1 + sin 2x
v<a;
2 tan 7°
(b) 2 sin 30 cos 30
\"i
2 tan 76»
1 - tan2 IB
21. (a)
(b) cos2 56 - sin:
ft
a
22. (a) cos2 — - sin2 —
(b) 2 sin — cos —
23. (a)
sin 8°
1 + cos 8°
e
(b)
n
cos
30°
in la.\2 34° -- sin
2 34°
IM
2 tan*
50. —r- = sin 2x
1 + tan2*
D 1.
sin 4x
sin*
52.
1 + sin 2*
sin 2*
53.
2(tan * — cot *)
tan2* — cot2*
I;R
<+^t
lllll *JJ\, —
e
1 - cos 46
sin 40
1
- cos 80
25-30 • Find sin —, cos —, and tan — from the given
information.
— «f cos * C( )S £X
1 - tan2*
2 tan*
3 tan * - tan3*
-,
1 - 3 tan2*
56. 4(sin6* + cos6*) = 4 - 3 sin2 2*
57. cos * - sin** = cos 2*
25. sin * = |, 0° < * < 90°
26. cos*= -f,
180°<*<270°
27. csc* = 3, 9 0 ° < * < 180°
29. sec * = f,
270° < * < 360°
30. cot* = 5,
180° < * < 2 7 0 °
31-34 • Write the product as a sum.
31. sin 2* cos 3*
32. sin* sin 5*
33. 3 cos 4* cos 7*
, . *
x
34. I I sin — cos —
2
4
sun
.1-st siiu
7T\ + Si
sin * + sin 5*
sin 3* + sin 7*
59. - = tan 3* 60. —
— = cot 2*
cos * + cos 5*
cos 3* — cos 7*
61.
28. tan* = 1, 0° < * < 90°
Jx
58. tan2 — + \
4/
.._
sin 10*
sin 9* + sin *
cos 5*
cos 4*
sin * + sin 3* + sin 5*
— = tan 3*
cos * + cos 3* cos 5*
62. 63.
sin * + sin y _
cos * + cos y
sm(* + y) — sm(* — y)
64. tan y = cos(* + y) + cos (* — y)
65. Show that sin 45° + sin 15° = sin 75°.
35-40 • Write the sum as a product.
66. Show that cos 87° + cos 33° = sin 63°.
35. sin 5* + sin 3*
36. sin * - sin 4*
67. Prove the identity
37. cos 4x — cos 6*
38. cos 9* + cos 2*
39. sin 2* — sin 7*
40. sin 3* + sin 4*
sin * + sin 2* + sin 3* + sin 4* + sin 5*
= tan 3x
cos * + cos 2* + cos 3* + cos 4* + cos 5*