Exercises: 8.2 Inverse Trigonometric Functions Exercises

Exercises for 8.2 Inverse Trigonometric Functions

Exercise Group

In Problems 1–4, which functions have an inverse function? Explain your answer.

1.
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Exercise Group

For Problems 5–8, graph the function and decide if it has an inverse function.

5.

[latex]f(x)=\sin 2x - \cos x[/latex]

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8.

Exercise Group

For Problems 9–14, use a calculator to evaluate. Round your answers to the nearest tenth of a degree.

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14.

Exercise Group

For Problems 15–20, give exact values in radians.

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20.

Exercise Group

For Problems 21–26, sketch a figure to help you model each problem.

21.

Delbert is watching the launch of a satellite at Cape Canaveral. The viewing area is 500 yards from the launch site. The angle of elevation, [latex]\theta[/latex] of Delbert’s line of sight increases as the booster rocket rises.

  1. Write a formula for the height, [latex]h[/latex] of the rocket as a function of [latex]\theta[/latex]
  2. Write a formula for [latex]\theta[/latex] as a function of [latex]h[/latex]
  3. Evaluate the formula in part (b) for [latex]h=1000[/latex] and interpret the result.
22.

Francine’s house lies under the flight path from the city airport, and commercial airliners pass overhead at an altitude of 35,000 feet. As Francine watches an airplane recede, its angle of elevation, [latex]\theta[/latex] decreases.

  1. Write a formula for the horizontal distance, [latex]d[/latex] to the airplane as a function of [latex]\theta[/latex]
  2. Write a formula for [latex]\theta[/latex] as a function of [latex]d[/latex]
  3. Evaluate the formula in part (b) for [latex]d=20,000[/latex] and interpret the result.
23.

While driving along the interstate, you approach an enormous 50-foot-wide billboard that sits just beside the road. Your viewing angle, [latex]\theta[/latex] increases as you get closer to the billboard.

  1. Write a formula for your distance, [latex]d[/latex] from the billboard as a function of [latex]\theta[/latex]
  2. Write a formula for [latex]\theta[/latex] as a function of [latex]d[/latex]
  3. Evaluate the formula in part (b) for [latex]d=200[/latex] and interpret the result.
24.

Emma is walking along the bank of a straight river toward a 20-meter-long bridge over the river. Let [latex]\theta[/latex] be the angle subtended horizontally by Emma’s view of the bridge.

  1. Write a formula for Emma’s distance from the bridge, [latex]d[/latex] as a function of [latex]\theta[/latex]
  2. Write a formula for [latex]\theta[/latex] as a function of [latex]d[/latex]
  3. Evaluate the formula in part (b) for [latex]d=500[/latex] and interpret the result.
25.

Martin is viewing a 4-meter-tall painting whose base is 1 meter above his eye level.

  1. Write a formula for [latex]\alpha[/latex] the angle subtended from Martin’s eye level to the bottom of the painting, when he stands [latex]x[/latex] meters from the wall.
  2. Write a formula for [latex]\beta[/latex] the angle subtended by the painting, in terms of [latex]x[/latex]
  3. Evaluate the formula in part (b) for [latex]x=5[/latex] and interpret the result.
26.

A 5-foot mirror is positioned so that its bottom is 1.5 feet below Jane’s eye level.

  1. Write a formula for [latex]\alpha[/latex] the angle subtended by the section of mirror below Jane’s eye level, when she stands [latex]x[/latex] feet from the mirror.
  2. Write a formula for [latex]\theta[/latex] the angle subtended by the entire mirror, in terms of [latex]x[/latex]
  3. Evaluate the formula in part (b) for [latex]x=10[/latex] and interpret the result.

Exercise Group

For Problems 27–32, solve the formula for the given variable.

27.

[latex]V=V_0 \sin(2\pi\omega t+\phi)[/latex] for [latex]t[/latex]

28.

[latex]R=\dfrac v_0^2\sin (2\theta)[/latex] for [latex]\theta[/latex]

29.
30.

[latex]c^2=a^2 + b^2 - 2ab\cos C[/latex] for [latex]C[/latex]

31.

[latex]P=\dfrac[/latex] for [latex]\theta[/latex]

32.

Exercise Group

For Problems 33–38, find exact values without using a calculator.