6.3.2: Homework

Concept Check

1. Provide an example to show that doubling an angle does not double its sine or cosine.

Basic Skills

For the following exercises, use the unit circle to estimate trigonometric values. Then verify with your calculator.

2. Does \(\sin \left(2 \cdot 90^{\circ}\right) =2 \sin \left(90^{\circ}\right)\)?

3. Does \(\cos \left(2 \cdot 25^{\circ}\right)=2 \cos \left(25^{\circ}\right)\)?

4. Does \(\tan \left(2 \cdot 70^{\circ}\right)=2 \tan \left(70^{\circ}\right)\)?

5. Does \(\tan \left(2 \cdot 100^{\circ}\right)=2 \tan \left(100^{\circ}\right)\)?

For the following exercises, verify that each statement is true

6. \(\sin \left(90^{\circ}\right)=2 \sin \left(45^{\circ}\right) \cos \left(45^{\circ}\right)\)

7. \(\sin \left(60^{\circ}\right)=2 \sin \left(30^{\circ}\right) \cos \left(30^{\circ}\right)\)

8. \(\cos \left(60^{\circ}\right)=\cos^2 \left(30^{\circ}\right)-\sin^2 \left(30^{\circ}\right)\)

9. \(\tan \left(60^{\circ}\right)=\dfrac{2 \tan \left(30^{\circ}\right)}{1-\tan^2 \left(30^{\circ}\right)}\)

For the following exercises, determine if the statement is true or false. Explain your answer.

10. If \(\cos \left(\alpha\right)=0.32\), then \(\cos \left(2 \alpha\right)=0.64\).

11. If \(\sin\left( \phi\right)=r\), then \(\sin \left(2 \phi\right)=2 r\).

For the following exercises, use the Double Angle Identities to simplify the expression. Leave your answer in exact form.

12. \(2 \sin \left(34^{\circ}\right) \cos \left(34^{\circ}\right)\)

13. \(\cos^2\left( \dfrac{\pi}{10}\right)-\sin^2 \left(\dfrac{\pi}{10}\right)\)

14. \(1-2 \sin^2 \left(\dfrac{\pi}{16}\right)\)

15. \(2 \cos^2 \left(18^{\circ}\right)-1\)

16. \(\cos^2 \left(3 \theta\right)-\sin^2 \left(3 \theta\right)\)

17. \(2 \sin \left(2 \alpha\right) \cos \left(2 \alpha\right)\)

18. \(2 \sin \left(5 t\right) \cos \left(5 t\right)\)

19. \(\cos^2 \left(4 w\right)-\sin^2 \left(4 w\right)\)

20. \(\dfrac{2 \tan \left(64^{\circ}\right)}{1-\tan^2\left( 64^{\circ}\right)}\)

21. \(\dfrac{2 \tan \left(\frac{\pi}{5}\right)}{1-\tan^2 \left(\frac{\pi}{5}\right)}\)

22. \(2 \cos^2 \left(2 \beta\right)-1\)

23. \(1-2 \sin^2 \left(6 s\right)\)

24. \(\cos ^{2}\left(28^{\circ}\right)-\sin^{2}\left(28^{\circ}\right)\)

25. \(2\cos^{2} \left(37^{\circ} \right)-1\)

26. \(1-2\sin^{2}\left(17^{\circ}\right)\)

27. \(\cos^{2}\left(37^{\circ} \right)-\sin ^{2}\left(37^{\circ}\right)\)

28. \(\cos^{2} \left(9x\right)-\sin^{2} \left(9x\right)\)

29. \(\cos ^{2} \left(6x\right)-\sin ^{2} \left(6x\right)\)

30. \(4\sin \left(8x\right) \cos\left(8x\right)\)

31. \(6\sin \left(5x\right) \cos\left(5x\right)\)

For the following exercises, use the figures to find the trigonometric ratios.

32.  

    1. \(\sin \left(\alpha\right)\)

    2. \(\cos \left(\alpha\right)\)

    3. \(\tan \left(\alpha\right)\)

    4. \(\sin \left(2\alpha\right)\)

    5. \(\cos \left(2\alpha\right)\)

    6. \(\tan \left(2\alpha\right)\)

33.  

    1. \(\sin \left(\beta\right)\)

    2. \(\cos \left(\beta\right)\)

    3. \(\tan \left(\beta\right)\)

    4. \(\sin \left(2\beta\right)\)

    5. \(\cos \left(2\beta\right)\)

    6. \(\tan \left(2\beta\right)\)

34.  

    1. \(\sin \left(s\right)\)

    2. \(\cos \left(s\right)\)

    3. \(\tan \left(s\right)\)

    4. \(\sin \left(2s\right)\)

    5. \(\cos \left(2s\right)\)

    6. \(\tan \left(2s\right)\)

35.  

    1. \(\sin \left(t\right)\)

    2. \(\cos \left(t\right)\)

    3. \(\tan \left(t\right)\)

    4. \(\sin \left(2t\right)\)

    5. \(\cos \left(2t\right)\)

    6. \(\tan \left(2t\right)\)

36. Suppose \(\cos \left(\theta\right)=\frac{12}{13}\) and \(\frac{3 \pi}{2}<\theta<2 \pi\). Compute exact values for the following.

    1. \(\sin \left(\theta\right)\)

    2. \(\sin \left(2 \theta\right)\)

    3. \(\cos \left(2 \theta\right)\)

    4. \(\tan \left(2 \theta\right)\)

    5. Sketch the angles \(\theta\) and \(2 \theta\).

37. Suppose \(\sin \left(\phi\right)=\frac{5}{6}\) and \(\frac{\pi}{2}<\phi<\pi\). Compute exact values for the following.

    1. \(\cos \left(\phi\right)\)

    2. \(\sin \left(2 \phi\right)\)

    3. \(\cos \left(2 \phi\right)\)

    4. \(\tan \left(2 \phi\right)\)

    5. Sketch the angles \(\phi\) and \(2 \phi\).

38. If \(\tan \left(u\right)=-4\) and \(270^{\circ}<u<360^{\circ}\), find exact values for the following.

    1. \(\tan \left(2 u\right)\)

    2. \(\cos \left(2 u\right)\)

    3. \(\sin \left(2 u\right)\)

39. If \(\tan \left(v\right)=\frac{2}{3}\) and \(180^{\circ}<v<270^{\circ}\), find exact values for the following.

    1. \(\tan \left(2v\right)\)

    2. \(\cos \left(2v\right)\)

    3. \(\sin \left(2v\right)\)

For the following exercises, use the given information about \(\theta\) to find the exact values of \(\sin(2\theta)\), \(\cos(2\theta)\), and \(\tan(2\theta)\).

40. \(\sin(\theta) = -\dfrac{7}{25}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

41. \(\cos(\theta) = \dfrac{28}{53}\) where \(0 < \theta < \dfrac{\pi}{2}\)

42. \(\tan(\theta) = \dfrac{12}{5}\) where \(\pi < \theta < \dfrac{3\pi}{2}\)

43. \(\csc(\theta) = 4\) where \(\dfrac{\pi}{2} < \theta < \pi\)

44. \(\cos(\theta) = \dfrac{3}{5}\) where \(0 < \theta < \dfrac{\pi}{2}\)

45. \(\sin(\theta) = -\dfrac{4}{5}\) where \(\pi < \theta < \dfrac{3\pi}{2}\)

46. \(\cos(\theta) = \dfrac{12}{13}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

47. \(\sin(\theta) = \dfrac{5}{13}\) where \(\dfrac{\pi}{2} < \theta < \pi\)

48. \(\sec(\theta) = \sqrt{5}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

49. \(\tan(\theta) = -2\) where \(\dfrac{\pi}{2} < \theta < \pi\)

50. Suppose \(\theta\) is a Quadrant I angle with \(\sin(\theta) = x\). Verify the following formulas

    1. \(\cos(\theta) = \sqrt{1-x^2}\)

    2. \(\sin(2\theta) = 2x\sqrt{1-x^2}\)

    3. \(\cos(2\theta) = 1 - 2x^2\)

51. Suppose \(\theta\) is a Quadrant I angle with \(\tan(\theta) = x\). Verify the following formulas

    1. \(\cos(\theta) = \dfrac{1}{\sqrt{x^2+1}}\)

    2. \(\sin(\theta) = \dfrac{x}{\sqrt{x^2+1}}\)

    3. \(\sin(2\theta) = \dfrac{2x}{x^2+1}\)

    4. \(\cos(2\theta) = \dfrac{1-x^2}{x^2+1}\)

52. If \(\sin(\theta) = \dfrac{x}{2}\) for \(-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}\), find an expression for \(\cos(2\theta)\) in terms of \(x\).

53. If \(\tan(\theta) = \dfrac{x}{7}\) for \(-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}\), find an expression for \(\sin(2\theta)\) in terms of \(x\).

For the following exercises, graph one cycle of the given function by hand.

54. \( f(x) = 4\sin\left( 3x \right)\cos\left( 3x \right) \)

55. \( g(x) = \dfrac{2}{3}\cos^2\left( \frac{\pi}{3} x \right) - \dfrac{1}{2} \)

56. \( h(x) = -\dfrac{5}{7} \sin\left( \frac{1}{3}x \right) \cos\left( \frac{1}{3} x \right) \)

57. \( k(x) = \dfrac{3 \tan\left( \frac{1}{2}x \right)}{2 - 2\tan^2\left( \frac{1}{2}x \right)} \)

58. \( m(x) = 2\cos^2\left( 5x \right) - 2\sin^2\left( 5x \right) \)

Proofs

For the following exercises, prove the given identity for any right triangle \( \triangle ABC \), where \( C \) is a right angle.

59. \(\sin\left(A-B\right) = \cos\left(2B\right)\)

60. \(\cos\left(A-B\right) = \sin\left(2A\right)\)

61. \(\sin\left(2A\right) = \dfrac{2ab}{c^2}\)

62. \(\cos\left(2A\right) = \dfrac{b^2 - a^2}{c^2}\)

63. \(\tan\left(2A\right) = \dfrac{2ab}{b^2 - a^2}\)

64. Prove the Double-Angle Identity \(\sin \left(2 \theta\right)=2 \sin\left( \theta\right) \cos \left(\theta\right)\).

65. Prove the Double-Angle Identity \(\cos \left(2 \theta\right)=\cos^2 \left(\theta\right)-\sin^2 \left(\theta\right)\).

For the following exercises,
(a) Use graphs to decide if the equation is an identity.
(b) If the equation is not an identity, find a value of the variable that makes the equation false.

66. \(\cos \left(4 \theta\right)=4 \cos \left(\theta\right)\)

67. \(\cos \left(2 x\right)=\left(\cos \left(x\right)+\sin \left(x\right)\right)\left(\cos\left( x\right)-\sin \left(x\right)\right)\)

For the following exercises, prove the identity. Assume all quantities are defined.

68. \((\cos(\theta) + \sin(\theta))^2 = 1 + \sin(2\theta)\)

69. \((\cos(\theta) - \sin(\theta))^2 = 1 - \sin(2\theta)\)

70. \( \dfrac{\sin\left( 2\theta \right)}{\sin\left( \theta \right)} - \dfrac{\cos\left( 2\theta \right)}{\cos\left( \theta \right)} = \sec\left( \theta \right) \)

71. \( \dfrac{\sin\left( 3\theta \right)}{\sin\left( \theta \right)} - \dfrac{\cos\left( 3\theta \right)}{\cos\left( \theta \right)} = 2 \)

72. \(\tan(2\theta) = \dfrac{1}{1-\tan(\theta)} - \dfrac{1}{1+\tan(\theta)}\)

73. \(\csc(2\theta) = \dfrac{\cot(\theta) + \tan(\theta)}{2}\)

74. \(\left(\sin ^{2} x-1\right)^{2} =\cos \left(2x\right)+\sin ^{4} x\)

75. \(\sin \left(2x\right)=\dfrac{2\tan \left(x\right)}{1+\tan ^{2} \left(x\right)}\)

76. \(\tan \left(2x\right)=\dfrac{2\sin \left(x\right)\cos \left(x\right)}{2\cos ^{2} \left(x\right)-1}\)

77. \(\cot \left(x\right)-\tan \left(x\right)=2\cot \left(2x\right)\)

78. \(\dfrac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)} =\tan \left(\theta \right)\)

79. \(\cos \left(2\alpha \right)=\dfrac{1-\tan ^{2} \left(\alpha \right)}{1+\tan ^{2} \left(\alpha \right)}\)

80. \(\dfrac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos \left(t\right)} =\dfrac{2\cos \left(t\right)}{2\sin \left(t\right)-1}\)

81. \(\sin \left(3x\right)=3\sin \left(x\right)\cos ^{2} \left(x\right)-\sin ^{3} (x)\)

82. \(\cos \left(3x\right)=\cos ^{3} (x)-3\sin ^{2} (x)\cos \left(x\right)\)

83. \(8 \sin^{4}(\theta) = \cos(4\theta) - 4\cos(2\theta)+3\)

84. \(8 \cos^{4}(\theta) = \cos(4\theta) + 4\cos(2\theta)+3\)

85. \(\sin(3\theta) = 3\sin(\theta) - 4\sin^{3}(\theta)\)

86. \(\sin(4\theta) = 4\sin(\theta)\cos^{3}(\theta) - 4\sin^{3}(\theta)\cos(\theta)\)

87. \(\cos(4\theta) = 8\cos^{4}(\theta) - 8\cos^{2}(\theta) + 1\)

88. \(\cos(8\theta) = 128\cos^{8}(\theta)-256\cos^{6}(\theta)+160\cos^{4}(\theta)-32\cos^{2}(\theta)+1\)

89. \(\sec(2\theta) = \dfrac{\cos(\theta)}{\cos(\theta) + \sin(\theta)} + \dfrac{\sin(\theta)}{\cos(\theta)-\sin(\theta)}\)

90. \(\dfrac{1}{\cos(\theta) - \sin(\theta)} + \dfrac{1}{\cos(\theta) + \sin(\theta)} = \dfrac{2\cos(\theta)}{\cos(2\theta)}\)

91. \(\dfrac{1}{\cos(\theta) - \sin(\theta)} - \dfrac{1}{\cos(\theta) + \sin(\theta)} = \dfrac{2\sin(\theta)}{\cos(2\theta)}\)

92. \( \dfrac{\cos^2\left( \psi \right)}{\cos^2\left( \theta \right)} = \dfrac{1 + \cos\left( 2 \psi \right)}{1 + \cos\left( 2 \theta \right)} \)

Challenge Problems

93. Some trigonometry textbooks used to claim (incorrectly) that \(\sin\left(\theta\right) + \cos\left(\theta\right) = \sqrt{1 + \sin\left(2\theta\right)} \) was an identity. Give an example of a specific angle \(\theta \) that would make that equation false. Is \(\sin\left(\theta\right) + \cos\left(\theta\right) = \pm\sqrt{1 + \sin\left(2\theta\right)} \) an identity? Justify your answer.

94. In general, what is the largest value that \(\sin\left(\theta\right)\cos\left(\theta\right) \) can take? Justify your answer.