Exercises: Chapter 8 Review Problems

1.

[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]

2.

[latex]\cos \left(\dfrac{\pi}{3} - t\right)=\dfrac{1}{2}-\cos t[/latex]

3.

[latex]\tan (z-w)=\dfrac{\sin (z-w)}{\cos (z-w)}[/latex]

4.

[latex]\sin 2\phi = 1 - \cos 2\phi[/latex]

5.

[latex]\sin \left(\dfrac{\pi}{2}-x\right)=1-\sin x[/latex]

6.

[latex]\sin (\pi - x)=\sin x[/latex]

7.

[latex]\cos^2 \alpha - \sin^2 \alpha = -1[/latex]

8.

[latex]\tan^{-1} s = \dfrac{1}{\tan s}[/latex]

9.

If [latex]\sin x = -0.4[/latex] and [latex]\cos x \gt 0,[/latex] find an exact value for [latex]\cos \left(x + \dfrac{3\pi}{4}\right)\text{.}[/latex]

10.

If [latex]\cos x = -0.75[/latex] and [latex]\sin x \lt 0,[/latex] find an exact value for [latex]\cos \left(x - \dfrac{4\pi}{3}\right)\text{.}[/latex]

11.

If [latex]\cos \theta = \dfrac{-3}{8},~ \pi \lt \theta \lt \dfrac{3\pi}{2},[/latex] and [latex]\sin \phi = \dfrac{1}{4},~ \dfrac{\pi}{2} \lt \phi \lt \pi,[/latex] find exact values for

1. [latex]\displaystyle \sin (\theta + \phi)[/latex]
2. [latex]\displaystyle \tan (\theta + \phi)[/latex]

12.

If [latex]\sin \rho = \dfrac{5}{6},~ \dfrac{\pi}{2} \lt \rho \lt \pi,[/latex] and [latex]\cos \mu = \dfrac{-1}{3},~ \dfrac{\pi}{2} \lt \mu \lt \pi,[/latex] find exact values for

1. [latex]\displaystyle \cos (\rho - \mu)[/latex]
2. [latex]\displaystyle \tan (\rho - \mu)[/latex]

13.

If [latex]\tan (x + y) =2[/latex] and [latex]\tan y = \dfrac{1}{3},[/latex] find [latex]\tan x\text{.}[/latex]

14.

If [latex]\tan (x - y) =\dfrac{1}{4}[/latex] and [latex]\tan x = 4,[/latex] find [latex]\tan y\text{.}[/latex]

15.

[latex]\tan \left(t - \dfrac{5\pi}{3}\right)[/latex]

16.

[latex]\cos \left(s+ \dfrac{7\pi}{4}\right)[/latex]

17.

1. [latex]\displaystyle \sin \theta[/latex]
2. [latex]\displaystyle \cos \theta[/latex]
3. [latex]\displaystyle \tan \theta[/latex]
4. [latex]\displaystyle \sin 2\theta[/latex]
5. [latex]\displaystyle \cos 2\theta[/latex]
6. [latex]\displaystyle \tan 2\theta[/latex]

18.

1. [latex]\displaystyle \sin \phi[/latex]
2. [latex]\displaystyle \cos \phi[/latex]
3. [latex]\displaystyle \tan \phi[/latex]
4. [latex]\displaystyle \sin 2\phi[/latex]
5. [latex]\displaystyle \cos 2\phi[/latex]
6. [latex]\displaystyle \tan 2\phi[/latex]

19.

[latex]\sin 4x \cos 5x + \cos 4x \sin 5x[/latex]

20.

[latex]\cos 3\beta \cos 1.5 - \sin 3\beta \sin 1.5[/latex]

21.

[latex]\dfrac{\tan 2\phi - \tan 2}{1 + \tan 2\phi \tan 2}[/latex]

22.

[latex]\dfrac{\tan \dfrac{5\pi}{9} - \tan \dfrac{2\pi}{9}}{1 + \tan \dfrac{5\pi}{9} \tan \dfrac{2\pi}{9}}[/latex]

23.

[latex]2\sin 4\theta \cos 4\theta[/latex]

24.

[latex]1-2\sin^2 3\phi[/latex]

25.

[latex]\cos 2\theta - \sin \theta = 1[/latex]

26.

[latex]\tan 2z + \tan z = 0[/latex]

27.

[latex]f(x) = 4x - x^3[/latex]

28.

[latex]g(x) = 5 + \sqrt[3]{x - 2}[/latex]

29.

1. [latex]\displaystyle \tan^{-1}(-\sqrt{3})[/latex]
2. [latex]\displaystyle \arccos \left(-\dfrac{1}{2}\right)[/latex]

30.

1. [latex]\displaystyle \arcsin (-1)[/latex]
2. [latex]\displaystyle \cos^{-1}(-1)[/latex]

31.

An IMAX movie screen is 52.8 feet high.

1. If your line of sight is level with the bottom of the screen, write an expression for the angle subtended by the screen when you sit [latex]x[/latex] feet away.
2. Evaluate your expression for [latex]x = 20[/latex] feet and for [latex]x = 100[/latex] feet.

32.

Rembrandt’s painting The Night Watch measures 13 feet high by 16 feet wide.

1. Write an expression for the angle subtended by the width of the painting if you sit [latex]d[/latex] feet back from the center of the painting.
2. Evaluate your expression for [latex]d = 10[/latex] feet and for [latex]d = 25[/latex] feet.

33.

[latex]v_y = v_0\sin \theta - gt[/latex]

34.

[latex]\Delta W = -q_0 E\cos (\pi - \theta)\Delta l[/latex]

35.

[latex]\cos\left[\tan^{-1}\left(\dfrac{-\sqrt{5}}{2}\right)\right][/latex]

36.

[latex]\tan\left[\sin^{-1}\left(\dfrac{2}{7}\right)\right][/latex]

37.

[latex]\sin(\cos^{-1}2t)[/latex]

38.

[latex]\tan(\cos^{-1}m)[/latex]

39.

Explain why one of the expressions [latex]\sin^{-1}x[/latex] or [latex]\sin^{-1}\left(\dfrac{1}{x}\right)[/latex] must be undefined.

40.

Does [latex]\sin^{-1}(-x) = -\sin^{-1}(x)\text{?}[/latex] Does [latex]\cos^{-1}(-x) = -\cos^{-1}(x)\text{?}[/latex]

41.

1. [latex]\displaystyle \csc 27°[/latex]
2. [latex]\displaystyle \sec 108°[/latex]
3. [latex]\displaystyle \cot 245°[/latex]

42.

1. [latex]\displaystyle \csc 5.3[/latex]
2. [latex]\displaystyle \cot 0.98[/latex]
3. [latex]\displaystyle \sec 2.17[/latex]

43.

44.

45.

46.

47.

48.

49.

[latex]6\cos \alpha = -5\text{,}[/latex] [latex]~ 180° \lt \alpha \lt 270°[/latex]

50.

[latex]4\sin \theta = 3,~ \theta[/latex] is obtuse

51.

52.

53.

54.

55.

[latex]2\sin \alpha - k = 0,~\dfrac{\pi}{2} \lt \alpha \lt \pi[/latex]

56.

[latex]h\cos \beta - 3 = 0,~\dfrac{3\pi}{2} \lt \beta \lt 2\pi[/latex]

57.

58.

59.

[latex]4\cot \dfrac{3\pi}{4} - \sec^2 \dfrac{\pi}{3}[/latex]

60.

[latex]\dfrac{1}{2}\csc \dfrac{2\pi}{3} + \tan^2 \dfrac{5\pi}{6}[/latex]

61.

[latex]\csc \dfrac{7\pi}{6}\cos \dfrac{5\pi}{4}[/latex]

62.

[latex]\sec \dfrac{7\pi}{4}\cot \dfrac{4\pi}{3}[/latex]

63.

[latex]3\csc \theta + 2 = 12[/latex]

64.

[latex]5\cot \theta + 15 = -3[/latex]

65.

The graph has vertical asymptotes at multiples of [latex]\pi\text{.}[/latex]

66.

The graph has a horizontal asymptote at [latex]\dfrac{\pi}{2}\text{.}[/latex]

67.

The function values are the reciprocals of [latex]y = \cos x\text{.}[/latex]

68.

The function is defined only for [latex]x[/latex]-values between [latex]-1[/latex] and [latex]1\text{,}[/latex] inclusive.

69.

None of the function values lie between [latex]-1[/latex] and [latex]1\text{.}[/latex]

70.

The graph includes the origin.

71.

[latex]f(x)=\tan x(\cos x - \cot x)[/latex]

72.

[latex]g(x)=\csc x - \cot x\cos x[/latex]

73.

[latex]G(x) = \sin x(\sec x - \csc x)[/latex]

74.

[latex]F(x) = \dfrac{1}{2}\left(\dfrac{\cos x}{1+\sin x} + \dfrac{1+ \sin x}{\cos x}\right)[/latex]

75.

[latex]1-\dfrac{\sin x}{\csc x}[/latex]

76.

[latex]\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}[/latex]

77.

[latex]\dfrac{2+\tan^2 B}{\sec^2 B} - 1[/latex]

78.

[latex]\dfrac{\csc t}{\tan t + \cot t}[/latex]

79.

[latex]\dfrac{\sqrt{16+x^2}}{x},~~x = 4\tan \theta[/latex]

80.

[latex]x\sqrt{4-x^2},~~x=2\sin \theta[/latex]

81.

[latex]\dfrac{x^2 - 3}{x},~~x=\sqrt{3}\sec \theta[/latex]

82.

[latex]\dfrac{x}{\sqrt{x^2+2}},~~x=\sqrt{2}\tan \theta[/latex]

83.

This problem outlines a geometric proof of difference of angles formula for tangent.

1. In the figure below left, [latex]\alpha=\angle ABC[/latex] and [latex]\beta = \angle DBC\text{.}[/latex] Write expressions in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for the sides [latex]AC,~DC,[/latex] and [latex]AD\text{.}[/latex]
2. In the figure above right, explain why [latex]\triangle ABC[/latex] is similar to [latex]\triangle FBE\text{.}[/latex]
3. Explain why [latex]\angle FDC = \alpha\text{.}[/latex]
4. Write an expression in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for side [latex]CF\text{.}[/latex]
5. Explain why [latex]\triangle FBE[/latex] is similar to [latex]\triangle ADE\text{.}[/latex]
6. Justify each equality in the statement
   [latex]\tan (\alpha - \beta) = \dfrac{DE}{BE} = \dfrac{AD}{BF} = \dfrac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta}[/latex]

84.

Let [latex]L_1[/latex] and [latex]L_2[/latex] be two lines with slopes [latex]m_1[/latex] and [latex]m_2\text{,}[/latex] respectively, and let [latex]\theta[/latex] be the acute angle formed between the two lines. Use an identity to show that

[latex]\tan \theta = \dfrac{m_2-m_1}{1+m_1m_2}[/latex]

85.

In the figure above, find the diameter of the circumscribing circle, the angle [latex]\alpha\text{,}[/latex] and the sides [latex]a[/latex] and [latex]b\text{.}[/latex]

86.

A triangle has one side of length 17 cm and the angle opposite is [latex]26°\text{.}[/latex] Find the diameter of the circle that circumscribes the triangle.