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Review Exercises

Determine whether the statement is true or false.

1. ${sin}^{2} s \neq sin s^{2}$ ${sin}^{2} s \neq sin s^{2}$ .[7.1]
2. Given $0 < α < π / 2$ $0 < α < π / 2$ and $0 < β < π / 2$ $0 < β < π / 2$ and that $sin (α + β) = 1$ $sin (α + β) = 1$ and $sin (α - β) = 0$ $sin (α - β) = 0$ , then $α = π / 4$ $α = π / 4$ .[7.1]
3. If the terminal side of $θ$ $θ$ is in quadrant IV, then $tan θ < cos θ$ $tan θ < cos θ$ .[7.1]
4. $cos 5 π / 12 = cos 7 π / 12$ $cos 5 π / 12 = cos 7 π / 12$ .[7.2]
5. Given that $sin θ = - \frac{2}{5}, tan θ < cos θ$ $sin θ = - \frac{2}{5}, tan θ < cos θ$ .[7.1]

Complete the Pythagorean identity. [7.1]

6. $1 + {cot}^{2} x =$ $1 + {cot}^{2} x =$
7. ${sin}^{2} x + {cos}^{2} x =$ ${sin}^{2} x + {cos}^{2} x =$

Multiply and simplify. [7.1]

8. $(tan y - cot y) (tan y + cot y)$ $(tan y - cot y) (tan y + cot y)$
9. ${(cos x + sec x)}^{2}$ ${(cos x + sec x)}^{2}$

Factor and simplify. [7.1]

10. $sec x csc x - {csc}^{2} x$ $sec x csc x - {csc}^{2} x$
11. $3 {sin}^{2} y - 7 sin y - 20$ $3 {sin}^{2} y - 7 sin y - 20$
12. $1000 - {cos}^{3} u$ $1000 - {cos}^{3} u$

Simplify. [7.1]

13. $\frac{{sec}^{4} x - {tan}^{4} x}{{sec}^{2} x + {tan}^{2} x}$ $\frac{{sec}^{4} x - {tan}^{4} x}{{sec}^{2} x + {tan}^{2} x}$
14. $\frac{2 {sin}^{2} x}{{cos}^{3} x} \cdot {(\frac{cos x}{2 sin x})}^{2}$ $\frac{2 {sin}^{2} x}{{cos}^{3} x} \cdot {(\frac{cos x}{2 sin x})}^{2}$
15. $\frac{3 sin x}{{cos}^{2} x} \cdot \frac{{cos}^{2} x + cos x sin x}{{sin}^{2} x - {cos}^{2} x}$ $\frac{3 sin x}{{cos}^{2} x} \cdot \frac{{cos}^{2} x + cos x sin x}{{sin}^{2} x - {cos}^{2} x}$
16. $\frac{3}{cos y - sin y} - \frac{2}{{sin}^{2} y - {cos}^{2} y}$ $\frac{3}{cos y - sin y} - \frac{2}{{sin}^{2} y - {cos}^{2} y}$
17. ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$ ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$
18. $\frac{4 sin x {cos}^{2} x}{16 {sin}^{2} x cos x}$ $\frac{4 sin x {cos}^{2} x}{16 {sin}^{2} x cos x}$

In Exercises 19–21, assume that all radicands are nonnegative.

19. Simplify:

$\sqrt{{sin}^{2} x + 2 cos x sin x + {cos}^{2} x}$ $\sqrt{{sin}^{2} x + 2 cos x sin x + {cos}^{2} x}$ . [7.1]
20. Rationalize the denominator: $\sqrt{\frac{1 + sin x}{1 - sin x}}$ $\sqrt{\frac{1 + sin x}{1 - sin x}}$ . [7.1]
21. Rationalize the numerator: $\sqrt{\frac{cos x}{tan x}}$ $\sqrt{\frac{cos x}{tan x}}$ . [7.1]
22. Given that $x = 3 tan θ$ $x = 3 tan θ$ , express $\sqrt{9 + x^{2}}$ $\sqrt{9 + x^{2}}$ as a trigonometric function without radicals. Assume that $0 < θ < π / 2$ $0 < θ < π / 2$ . [7.1]

Use the sum and difference formulas to write equivalent expressions. You need not simplify. [7.1]

23. $cos (x + \frac{3 π}{2})$ $cos (x + \frac{3 π}{2})$
24. $tan (45 ° - 30 °)$ $tan (45 ° - 30 °)$
25. Simplify: $cos 27 ° cos 16 ° + sin 27 ° sin 16 °$ $cos 27 ° cos 16 ° + sin 27 ° sin 16 °$ . [7.1]
26. Find $cos 165 °$ $cos 165 °$ exactly. [7.1]
27. Given that $tan α = \sqrt{3}$ $tan α = \sqrt{3}$ and $sin β = \sqrt{2} / 2$ $sin β = \sqrt{2} / 2$ and that $α$ $α$ and $β$ $β$ are between 0 and $π / 2$ $π / 2$ , evaluate $\tan (α - β)$ $\tan (α - β)$ exactly. [7.1]
28. Assume that $sin θ = 0.5812$ $sin θ = 0.5812$ and $cos ϕ = 0.2341$ $cos ϕ = 0.2341$ and that both $θ$ $θ$ and $ϕ$ $ϕ$ are first-quadrant angles. Evaluate $cos (θ + ϕ)$ $cos (θ + ϕ)$ . [7.1]

Complete the cofunction identity. [7.2]

29. $cos (x + \frac{π}{2}) =$ $cos (x + \frac{π}{2}) =$
30. $cos (\frac{π}{2} - x) =$ $cos (\frac{π}{2} - x) =$
31. $sin (x - \frac{π}{2}) =$ $sin (x - \frac{π}{2}) =$
32. Given that $cos α = - \frac{3}{5}$ $cos α = - \frac{3}{5}$ and that the terminal side is in quadrant III:
1. a) Find the other function values for $α$ $α$ . [7.2]
2. b) Find the six function values for $π / 2 - α$ $π / 2 - α$ . [7.2]
3. c) Find the six function values for $α + π / 2$ $α + π / 2$ . [7.2]
33. Find an equivalent expression for $csc (x - \frac{π}{2})$ $csc (x - \frac{π}{2})$ . [7.2]
34. Find $tan 2 θ, cos 2 θ$ $tan 2 θ, cos 2 θ$ , and $sin 2 θ$ $sin 2 θ$ and the quadrant in which $2 θ$ $2 θ$ lies, where $cos θ = - \frac{4}{5}$ $cos θ = - \frac{4}{5}$ and $θ$ $θ$ is in quadrant III. [7.2]
35. Find $sin \frac{π}{8}$ $sin \frac{π}{8}$ exactly. [7.2]
36. Given that $sin β = 0.2183$ $sin β = 0.2183$ and $β$ $β$ is in quadrant I, find $sin 2 β, cos \frac{β}{2}$ $sin 2 β, cos \frac{β}{2}$ , and $cos 4 β$ $cos 4 β$ . [7.2]

Simplify. [7.2]

37. $1 - 2 {sin}^{2} \frac{x}{2}$ $1 - 2 {sin}^{2} \frac{x}{2}$
38. ${(sin x + cos x)}^{2} - sin 2 x$ ${(sin x + cos x)}^{2} - sin 2 x$
39. $2 sin x {cos}^{3} x + 2 {sin}^{3} x cos x$ $2 sin x {cos}^{3} x + 2 {sin}^{3} x cos x$
40. $\frac{2 cot x}{{cot}^{2} x - 1}$ $\frac{2 cot x}{{cot}^{2} x - 1}$

Prove the identity. [7.3]

41. $\frac{1 - sin x}{cos x} = \frac{cos x}{1 + sin x}$ $\frac{1 - sin x}{cos x} = \frac{cos x}{1 + sin x}$
42. $\frac{1 + cos 2 θ}{sin 2 θ} = cot θ$ $\frac{1 + cos 2 θ}{sin 2 θ} = cot θ$
43. $\frac{tan y + sin y}{2 tan y} = {cos}^{2} \frac{y}{2}$ $\frac{tan y + sin y}{2 tan y} = {cos}^{2} \frac{y}{2}$
44. $\frac{sin x - cos x}{{cos}^{2} x} = \frac{{tan}^{2} x - 1}{sin x + cos x}$ $\frac{sin x - cos x}{{cos}^{2} x} = \frac{{tan}^{2} x - 1}{sin x + cos x}$

Use the product-to-sum identities and the sum-to-product identities to find identities for each of the following. [7.3]

45. $3 cos 2 θ sin θ$ $3 cos 2 θ sin θ$
46. $sin θ - s i n 4 θ$ $sin θ - s i n 4 θ$

Find each of the following exactly in both radians and degrees. [7.4]

47. ${sin}^{- 1} (- \frac{1}{2})$ ${sin}^{- 1} (- \frac{1}{2})$
48. ${cos}^{- 1} \frac{\sqrt{3}}{2}$ ${cos}^{- 1} \frac{\sqrt{3}}{2}$
49. ${tan}^{- 1} 1$ ${tan}^{- 1} 1$
50. ${sin}^{- 1} 0$ ${sin}^{- 1} 0$

Use a calculator to find each of the following both in radians, rounded to four decimal places, and in degrees, rounded to the nearest tenth of a degree. [7.4]

51. ${cos}^{- 1} (- 0.2194)$ ${cos}^{- 1} (- 0.2194)$
52. ${cot}^{- 1} 2.381$ ${cot}^{- 1} 2.381$

Evaluate. [7.4]

53. $cos ({cos}^{- 1} \frac{1}{2})$ $cos ({cos}^{- 1} \frac{1}{2})$
54. ${tan}^{- 1} (tan \frac{\sqrt{3}}{3})$ ${tan}^{- 1} (tan \frac{\sqrt{3}}{3})$
55. ${sin}^{- 1} (sin \frac{π}{7})$ ${sin}^{- 1} (sin \frac{π}{7})$
56. $cos ({sin}^{- 1} \frac{\sqrt{2}}{2})$ $cos ({sin}^{- 1} \frac{\sqrt{2}}{2})$

Find each of the following. [7.4]

57. $cos ({tan}^{- 1} \frac{b}{3})$ $cos ({tan}^{- 1} \frac{b}{3})$
58. $cos (2 {sin}^{- 1} \frac{4}{5})$ $cos (2 {sin}^{- 1} \frac{4}{5})$

Solve, finding all solutions. Express the solutions in both radians and degrees. [7.5]

59. $cos x = - \frac{\sqrt{2}}{2}$ $cos x = - \frac{\sqrt{2}}{2}$
60. $tan x = \sqrt{3}$ $tan x = \sqrt{3}$

Solve, finding all solutions in $[0, 2 π)$ $[0, 2 π)$ . [7.5]

61. $4 {sin}^{2} x = 1$ $4 {sin}^{2} x = 1$
62. $sin 2 x sin x - cos x = 0$ $sin 2 x sin x - cos x = 0$
63. $2 {cos}^{2} x + 3 cos x = - 1$ $2 {cos}^{2} x + 3 cos x = - 1$
64. ${sin}^{2} x - 7 sin x = 0$ ${sin}^{2} x - 7 sin x = 0$
65. ${csc}^{2} x - 2 {cot}^{2} x = 0$ ${csc}^{2} x - 2 {cot}^{2} x = 0$
66. $sin 4 x + 2 sin 2 x = 0$ $sin 4 x + 2 sin 2 x = 0$
67. $2 cos x + 2 sin x = \sqrt{2}$ $2 cos x + 2 sin x = \sqrt{2}$
68. $6 {tan}^{2} x = 5 tan x + {sec}^{2} x$ $6 {tan}^{2} x = 5 tan x + {sec}^{2} x$
69. Which of the following is the domain of the function ${cos}^{- 1} x ?$ ${cos}^{- 1} x ?$ [7.4]
1. $(0, π)$ $(0, π)$
2. $[- 1, 1]$ $[- 1, 1]$
3. $[- π / 2, π / 2]$ $[- π / 2, π / 2]$
4. $(- \infty, \infty)$ $(- \infty, \infty)$
70. simplify: ${sin}^{- 1} (sin \frac{7 π}{6})$ ${sin}^{- 1} (sin \frac{7 π}{6})$ . [7.4]
1. $- π / 6$ $- π / 6$
2. $7 π / 6$ $7 π / 6$
3. $- 1 / 2$ $- 1 / 2$
4. $11 π / 6$ $11 π / 6$
71. The graph of $f (x) = {sin}^{- 1} x$ $f (x) = {sin}^{- 1} x$ is which of the following?[7.4]

Synthesis

72. Find the measure of the angle from $l_{1}$ $l_{1}$ to $l_{2} :$ $l_{2} :$

$l_{1} : x + y = 3 l_{2} : 2 x - y = 5.$ $l_{1} : x + y = 3 l_{2} : 2 x - y = 5.$

[7.1]
73. Find an identity for $cos (u + v)$ $cos (u + v)$ involving only cosines. [7.1], [7.2]
74. simplify: $cos (\frac{π}{2} - x) [csc x - sin x]$ $cos (\frac{π}{2} - x) [csc x - sin x]$ . [7.2]
75. Find $sin θ, cos θ$ $sin θ, cos θ$ , and $tan θ$ $tan θ$ under the given conditions:

$sin 2 θ = \frac{1}{5}, \frac{π}{2} \leq 2 θ < π .$ $sin 2 θ = \frac{1}{5}, \frac{π}{2} \leq 2 θ < π .$

[7.2]
76. Show that

${tan}^{- 1} x = \frac{{sin}^{- 1} x}{{cos}^{- 1} x}$ ${tan}^{- 1} x = \frac{{sin}^{- 1} x}{{cos}^{- 1} x}$

is not an identity. [7.4]
77. Solve $e^{cos x} = 1$ $e^{cos x} = 1$ in $[0, 2 π)$ $[0, 2 π)$ . [7.5]

Collaborative Discussion and Writing

78. Why are the ranges of the inverse trigonometric functions restricted? [7.4]
79. Miles lists his answer to a problem as $π / 6 + k π$ $π / 6 + k π$ , for any integer $k$ $k$ , while Jaylen lists his answer as $π / 6 + 2 k π$ $π / 6 + 2 k π$ and $7 π / 6 + 2 k π$ $7 π / 6 + 2 k π$ , for any integer $k$ $k$ . Are their answers equivalent? Why or why not? [7.5]
80. How does the graph of $y = {sin}^{- 1} x$ $y = {sin}^{- 1} x$ differ from the graph of $y = sin x ?$ $y = sin x ?$ [7.4]
81. What is the difference between a trigonometric equation that is an identity and a trigonometric equation that is not an identity? Give an example of each. [7.1], [7.5]
82. Why is it that

$sin \frac{5 π}{6} = \frac{1}{2}, but {sin}^{- 1} (\frac{1}{2}) \neq \frac{5 π}{6} ?$ $sin \frac{5 π}{6} = \frac{1}{2}, but {sin}^{- 1} (\frac{1}{2}) \neq \frac{5 π}{6} ?$

[7.4]

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Table of Contents for Review Exercises

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Table of Contents for
Review Exercises