Exercises: 8.3 The Reciprocal Functions Exercises

1.

[latex]\csc 27°[/latex]

2.

[latex]\sec 8°[/latex]

3.

[latex]\cot 65°[/latex]

4.

[latex]\csc 11°[/latex]

5.

[latex]\sec 1.4[/latex]

6.

[latex]\cot 4.3[/latex]

7.

[latex]\csc \dfrac{5\pi}{16}[/latex]

8.

[latex]\sec \dfrac{7\pi}{20}[/latex]

9.

[latex]\csc 30°[/latex]

10.

[latex]\sec 0°[/latex]

11.

[latex]\cot 45[/latex]

12.

[latex]\csc 60°[/latex]

13.

[latex]\sec 150°[/latex]

14.

[latex]\cot 120°[/latex]

15.

[latex]\csc 135°[/latex]

16.

[latex]\sec 270°[/latex]

17.

18.

19.

Evaluate. Round answers to three decimal places.

1. [latex]\displaystyle \cos 0.2[/latex]
2. [latex]\displaystyle (\cos 0.2)^{-1}[/latex]
3. [latex]\displaystyle \cos^{-1}0.2[/latex]
4. [latex]\displaystyle \dfrac{1}{\cos 0.2}[/latex]
5. [latex]\displaystyle \cos\dfrac{1}{0.2}[/latex]
6. [latex]\displaystyle \sec 0.2[/latex]

20.

Evaluate. Round answers to three decimal places.

1. [latex]\displaystyle \tan 3.2[/latex]
2. [latex]\displaystyle \tan^{-1} 3.2[/latex]
3. [latex]\displaystyle \cot 3.2[/latex]
4. [latex]\displaystyle \dfrac{1}{\tan 3.2}[/latex]
5. [latex]\displaystyle \tan \dfrac{1}{3.2}[/latex]
6. [latex]\displaystyle (\tan 3.2)^{-1}[/latex]

21.

22.

23.

24.

25.

26.

27.

28.

29.

The distance that sunlight must travel to pass through a layer of Earth’s atmosphere depends on both the thickness of the atmosphere and the angle of the sun.

1. Write an expression for the distance, [latex]d{,}[/latex] that sunlight travels through a layer of atmosphere of thickness [latex]h{.}[/latex]
2. Find the distance (to the nearest mile) that sunlight travels through a 100-mile layer of atmosphere when the sun is [latex]40°[/latex] above the horizon.

30.

In railroad design, the degree of curvature of a section of track is the angle subtended by a chord 100 feet long.

1. Use the figure to write an expression for the radius, [latex]r{,}[/latex] of a curve whose degree of curvature is [latex]\theta{.}[/latex] (Hint: The bisector of the angle [latex]\theta[/latex] is perpendicular to the chord.)
2. Find the radius of a curve whose degree of curvature is [latex]43°{.}[/latex]

31.

When a plane is tilted by an angle [latex]\theta[/latex] from the horizontal, the time required for a ball starting from rest to roll a horizontal distance of [latex]l[/latex] feet on the plane is

[latex]t=\sqrt{\dfrac{l}{8}\csc(2\theta)}~~ {seconds}[/latex]

1. How long, to the nearest 0.01 second, will it take the ball to roll 2 feet horizontally on a plane tilted by [latex]12°{?}[/latex]
2. Solve the formula for [latex]l[/latex] in terms of [latex]t[/latex] and [latex]\theta{.}[/latex]

32.

After a heavy rainfall, the depth, [latex]D{,}[/latex] of the runoff flow at a distance [latex]x[/latex] feet from the watershed down a slope at angle [latex]\alpha[/latex] is given by

[latex]D=(kx)^{0.6}(\cot \alpha)^{0.3}~~{inches}[/latex]

where [latex]k[/latex] is a constant determined by the surface roughness and the intensity of the runoff.

1. How deep, to the nearest 0.01 inch, is the runoff 100 feet down a slope of [latex]10°[/latex] if [latex]k=0.0006{?}[/latex]
2. Solve the formula for [latex]x[/latex] in terms of [latex]D[/latex] and [latex]\alpha{.}[/latex]

33.

34.

35.

36.

37.

38.

39.

The diagram shows a unit circle. Find six line segments whose lengths are, respectively, [latex]\sin t,~ \cos t,~ \tan t,~ \sec t,~ \csc t,[/latex] and [latex]\cot t{.}[/latex]

40.

Use the figure in Problem 39 to find each area in terms of the angle [latex]t{.}[/latex]

1. [latex]\displaystyle \triangle OAC[/latex]
2. [latex]\displaystyle \triangle OBD[/latex]
3. sector [latex]OAC[/latex]
4. [latex]\displaystyle \triangle OFE[/latex]

41.

[latex]\sec \theta = 2,~~\theta[/latex] in Quadrant IV

42.

[latex]\csc \phi = 4,~~\phi[/latex] in Quadrant II

43.

[latex]\csc \alpha = 3,~~\alpha[/latex] in Quadrant I

44.

[latex]\sec \beta = 4,~~\beta[/latex] in Quadrant IV

45.

[latex]\cot \gamma = \dfrac{1}{4},~~\gamma[/latex] in Quadrant III

46.

[latex]\tan \theta = 6,~~\theta[/latex] in Quadrant I

47.

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

48.

[latex]\dfrac{1}{2} \csc \dfrac{\pi}{6} - \dfrac{1}{4}\cot \dfrac{\pi}{6}[/latex]

49.

[latex]\dfrac{1}{2} \csc \dfrac{5\pi}{3}\cot \dfrac{3\pi}{4}[/latex]

50.

[latex]6 \cot \dfrac{7\pi}{6} \sec \dfrac{5\pi}{4}[/latex]

51.

[latex]\left(\csc \dfrac{2\pi}{3} - \sec \dfrac{3\pi}{4}\right)^2[/latex]

52.

[latex]\sec^2 \dfrac{5\pi}{6} \csc^2 \dfrac{4\pi}{3}[/latex]

53.

Complete the table and sketch a graph of [latex]y=\sec x{.}[/latex]

54.

Complete the table and sketch a graph of [latex]y=\csc x{.}[/latex]

55.

Use the graph of [latex]y=\sin x[/latex] to sketch a graph of its reciprocal, [latex]y=\csc x{.}[/latex]

56.

Use the graph of [latex]y=\cos x[/latex] to sketch a graph of its reciprocal, [latex]y=\sec x{.}[/latex]

57.

Complete the table and sketch a graph of [latex]y=\cot x{.}[/latex]

58.

Use the graphs of [latex]y=\cos x[/latex] and [latex]y=\sin x[/latex] to sketch a graph of [latex]y=\cot x = \dfrac{\cos x}{\sin x}{.}[/latex]

59.

[latex]y=\dfrac{\csc x}{\cot x}[/latex]

60.

[latex]y=\dfrac{\sec x}{\tan x}[/latex]

61.

[latex]y=\dfrac{\sec x \cot x}{\csc x}[/latex]

62.

[latex]y=\dfrac{\csc x \tan x}{\sec x}[/latex]

63.

[latex]y=\tan x \csc x[/latex]

64.

[latex]y=\sin x \sec x[/latex]

65.

[latex]3\csc \theta + 2=8[/latex]

66.

[latex]-2\sec \theta + 7=3[/latex]

67.

[latex]\sqrt{2}\sec \theta =-2[/latex]

68.

[latex]8+\csc \theta =6[/latex]

69.

[latex]2\cot \theta = -\sqrt{12}[/latex]

70.

[latex]\sqrt{3}\cot \theta =1[/latex]

71.

If [latex]\tan \alpha = -2[/latex] and [latex]\dfrac{\pi}{2} \lt \alpha \lt \pi{,}[/latex] find [latex]\cos \alpha{.}[/latex]

72.

If [latex]\cot \beta = \dfrac{5}{4}[/latex] and [latex]\pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find [latex]\sin \beta{.}[/latex]

73.

If [latex]\sec x = \dfrac{a}{2}[/latex] and [latex]0 \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\tan x{.}[/latex]

74.

If [latex]\csc y = \dfrac{1}{b}[/latex] and [latex]\dfrac{\pi}{2} \lt y \lt \pi{,}[/latex] find [latex]\cot y{.}[/latex]

75.

If [latex]\csc \phi = w[/latex] and [latex]\dfrac{3\pi}{2} \lt \alpha \lt 2\pi{,}[/latex] find [latex]\cos \phi{.}[/latex]

76.

If [latex]\sec \theta = \dfrac{3}{z}[/latex] and [latex]\pi \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\sin \theta{.}[/latex]

77.

78.

79.

80.

81.

[latex]\sec \theta \tan \theta[/latex]

82.

[latex]\csc \phi \cot \phi[/latex]

83.

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

84.

[latex]\dfrac{\tan v}{\sec v}[/latex]

85.

[latex]\sec \beta - \tan \beta[/latex]

86.

[latex]\cot \alpha + \csc \alpha[/latex]

87.

[latex]\sin x \tan x - \sec x[/latex]

88.

[latex]\csc y - \cos y \cot y[/latex]

89.

Prove the Pythagorean identity [latex]1 + \tan^2 \theta = \sec^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\cos^2 \theta{.}[/latex])

90.

Prove the Pythagorean identity [latex]1 + \cot^2 \theta = \csc^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\sin^2 \theta{.}[/latex])

91.

Suppose that [latex]\cot \theta = 5[/latex] and [latex]\theta[/latex] lies in the third quadrant.

1. Use the Pythagorean identity to find the value of [latex]\csc \theta{.}[/latex]
2. Use identities to find the values of the other four trig functions of [latex]\theta{.}[/latex]

92.

Suppose that [latex]\tan \theta = -2[/latex] and [latex]\theta[/latex] lies in the second quadrant.

1. Use the Pythagorean identity to find the value of [latex]\sec \theta{.}[/latex]
2. Use identities to find the values of the other four trig functions of [latex]\theta{.}[/latex]

93.

Write each of the other five trig functions in terms of [latex]\sin t[/latex] only.

94.

Write each of the other five trig functions in terms of [latex]\cos t[/latex] only.

95.

Show that if the angles of a triangle are [latex]A,~B,[/latex] and [latex]C[/latex] and the opposite sides are respectively [latex]a,~b,[/latex] and [latex]c,[/latex] then

[latex]a \csc A = b \csc B = c \csc C[/latex]

96.

The figure shows a unit circle and an angle [latex]\theta[/latex] in standard position. Each of the six trigonometric ratios for [latex]\theta[/latex] is represented by the length of a line segment in the figure. Find the line segment for each ratio and explain your choice.