5.3 Homework

4 4
~ v'2
212 I CHAPTER 5 Trigonometric Identities
l~j Exercises
-1
1.F 4. 2. •16. 6. BA
C-I4
10. 12. 14. 8 v6+v'2 v'2 v6 v'2 ~ v6 v'2 v6
11. v'2 ~ v6
(x + 3. 2. 6. 4. 5. y) cas('~-x) sin sin(~-x) cas(x = ___ (x - -~) y) ~ ) = = ___ __
___
17. The calculator gives a value of
o for the expression.
18. The calculator gives a value of
-I for the expression.
19. cot 3°
21 . 51T
· sm IT
23. sec 75° 36'
24. cos(-52° 14')
25. cos( -*)
26. tan( - 2;)
27. csc( -56° 42')
28. cot(-84°03')
29. tan( -86.9814°)
30. cos( -8.0142°)
31. tan
33. cos
35. csc
28. tan 174° 03'
8
20. cos 75°
22. cos TIT
32. cos
34. tan
Concept Check Match each expression in Column I with the correct expression in Column
II to form an identity. Choices may be used once, more than once, or not at all.
I II
A. D. C. B. E. F. cas Sill -cas -cas cas Sill X x Xcas
xcas
y + - sin x sin y
Use identities to find each exact value. (Do not use a calculator.) See Example 1.
7. cas 75°
2.: cas( -105°)
(Hint: -105° = -60° + (-45°»)
77T
11. cas 12
13. cas( - ~)
15. cas 40° cas 50° - sin 40° sin 50°
10. cas 105°
(Hint: 105° = 60° + 45°)
12. cas( ~)
14. cas( - ~~)
77T 27T 77T 27T
16. cas-cas-- sin-sin-
9 9 9 9
~ Use a graphing or scientific calculator to support your answer for each of thefollowing.
See Example 1.
36. tan 17. Exercise 15 18. Exercise 16
Write each function value in terms of the cofunction of a complementary angle. See
Example 2.
19. tan 87°
12
5
57T
10 20. 97T 26. 29. 30. 21. 24. cat- sin cat sin cas sin15° 176.9814° 98.0142° 142° ~ 14'
23. 27. csc(14° sec 146° 24') 42'
Use identities to fill in each blank with the appropriate trigonometric function name. See
Example 2.
7T
7T _
31. cat 3= -- 6
33. 33° = sin 57°
35. cas 70° = __ 200
32 sin - = - -
. 27T 3 -- ( 7T) 6
34. 72° = cat 18°
I
36. tan 24° ==- 66
- - '.•.
,,~~

For Exercises 37-42, other answers
are possible. We give the most
obvious one.
38. 20°
37. ISO
39. IjOO
41. 20°
43. cas 8
45. -cas 8
47. cas 8
49. -cas 8
40. 3~8°
42. 40°
44. sin 8
46. -sin 8
48. -sin 8
50. sin 8
51 lQ. -~ 52 _lIi.!i1
• 65' 65 • 85 ' 85
53 4 - 6Y6. 4 + 6Y6
• 25 ' 25
54 -2V1O + 2. -2V1O - 2
• 9 ' 9
55 2 \.1638 - v30. 2 \.1638 + v30
• 56 ' 56
56 V62 - v70. V62 + v70
• 24 24
57. true
59. false
61. true
63. true
65. false
58. false
60. true
62. true
64. true
66. false
SECTION5.3 Sum and Difference Identities for Cosine I 213
Find one angle 8 that satisfies each of the following. See Example 2.
37. tan 8 = cot( 45° + 28)
39. sec 8 = csc( ~ + 20°)
41. sin(38 - 15°) = cos(8 + 25°)
38. sin 8 = cos(28 + 30°)
40. cas 8 = sin( ~ + 3°)
Use the identities for the cosine of a sum or difference to write each expression as a
function of8. See Example 3.
43. cos(O° - 8)
45. cos(8 - 180°)
47. cos(O° + 8)
49. cos( 180° + 8)
Find cos(s + t) and cos(s - t). See Example 4.
51. sin s = ~and sin t = - H, s in quadrant I and t in quadrant III
52. cas s = - f? and cas t = - ~' sand t in quadrant III
53. cas s = - k and sin t = ~, sand t in quadrant II
54. sin s = ~and sin t = - t, s in quadrant II and t in quadrant IV
55' VSd' v6 d" d I
. sm s = -7- an sm t = -8-' S an t m qua rant
44. cos(90° - 8)
46. cos( 8 - 270°)
48. cos(90° + 8)
50. cos(270° + 8)
56. cas s =1and sin t = - f, sand t in quadrant IV
Concept Check Determine whether each statement is true or false.
58. cos( -24°) = cas 16° - cas 40°
59. cas 74° = cas 60° cas 14° + sin 60° sin 14°
60. cas 140° = cas 60° cas 80° - sin 60° sin 80°
7T 7T 7T 7T 7T
61. cas - = cas - cas - - sin - sin ­
- 3 12 4 12 4
27T 117T 7T 117T 7T
62. cas - = cas -- cas - + sin -- sin -
3 12 4 12 4
63. cas 70° cas 20° - sin 70° sin 20° = 0
64. cas 85° cas 40° + sin 85° sin 400 = v'2 2
65. tan(x - %) = cot x 66. sin (x - %) = cas x
Verify that each equation is an identity. See Example 6.
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