## Problem Set 50: Inverse Trigonometric Functions

1. Why do the functions $f(x)=\sin^{−1}x$ and $g(x)=\cos^{−1}x$ have different ranges?

2. Since the functions $y=\cos x$ and $y=\cos^{−1}x$ are inverse functions, why is $\cos^{−1}\left(\cos\left(−\frac{\pi}{6}\right)\right)$ not equal to $−\frac{\pi}{6}$?

3. Explain the meaning of $\frac{\pi}{6}=\arcsin(0.5)$.

4. Most calculators do not have a key to evaluate $\sec^{−1}(2)$. Explain how this can be done using the cosine function or the inverse cosine function.

5. Why must the domain of the sine function, $\sin x$, be restricted to $\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right]$ for the inverse sine function to exist?

6. Discuss why this statement is incorrect: $\arccos(\cos x)=x$ for all x.

7. Determine whether the following statement is true or false and explain your answer: $\arccos(−x)=\pi−\arccos x$.

For the following exercises, evaluate the expressions.

8. $\sin^{−1}\left(\frac{\sqrt{2}}{2}\right)$

9. $\sin^{−1}\left(−\frac{1}{2}\right)$

10. $\cos^{−1}\left(\frac{1}{2}\right)$

11. $\cos^{−1}\left(−\frac{\sqrt{2}}{2}\right)$

12. $\tan^{−1}(1)$

13. $\tan^{−1}(−\sqrt{3})$

14. $\tan^{−1}(−1)$

15. $\tan^{−1}(\sqrt{3})$

16. $\tan^{−1}\left(\frac{−1}{\sqrt{3}}\right)$

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

17. $\cos^{−1}(−0.4)$

18. $\arcsin(0.23)$

19. $\arccos\left(\frac{3}{5}\right)$

20. $\cos^{−1}(0.8)$

21. $\tan^{−1}(6)$

For the following exercises, find the angle θ in the given right triangle. Round answers to the nearest hundredth.

22.

23.

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

24. $\sin^{−1}(\cos(\pi))$

25. $\tan^{−1}(\sin(\pi))$

26. $\cos^{−1}\left(\sin\left(\frac{\pi}{3}\right)\right)$

27. $\tan^{−1}\left(\sin\left(\frac{\pi}{3}\right)\right)$

28. $\sin^{−1}\left(\cos\left(\frac{−\pi}{2}\right)\right)$

29. $\tan^{−1}\left(\sin\left(\frac{4\pi}{3}\right)\right)$

30. $\sin^{−1}\left(\sin\left(\frac{5\pi}{6}\right)\right)$

31. $\tan^{−1}\left(\sin\left(\frac{−5\pi}{2}\right)\right)$

32. $\cos\left(\sin^{−1}\left(\frac{4}{5}\right)\right)$

33. $\sin\left(\cos^{−1}\left(\frac{3}{5}\right)\right)$

34. $\sin\left(\tan^{−1}\left(\frac{4}{3}\right)\right)$

35. $\cos\left(\tan^{−1}\left(\frac{12}{5}\right)\right)$

36. $\cos\left(\sin^{−1}\left(\frac{1}{2}\right)\right)$

For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.

37. $\tan\left(\sin^{−1}\left(x−1\right)\right)$

38. $\sin\left(\cos^{−1}\left(1−x\right)\right)$

39. $\cos\left(\sin^{−1}\left(\frac{1}{x}\right)\right)$

40. $\cos\left(\tan^{−1}\left(3x−1\right)\right)$

41. $\tan\left(\sin^{−1}\left(x+\frac{1}{2}\right)\right)$

For the following exercises, evaluate the expression without using a calculator. Give the exact value.

42. $\frac{\sin^{−1}\left(\frac{1}{2}\right)−\cos^{−1}\left(\frac{\sqrt{2}}{2}\right)+\sin^{−1}\left(\frac{\sqrt{3}}{2}\right)−\cos^{−1}\left(1\right)}{\cos^{−1}\left(\frac{\sqrt{3}}{2}\right)−\sin^{−1}\left(\frac{\sqrt{2}}{2}\right)+\cos^{−1}\left(\frac{1}{2}\right)−\sin^{−1}\left(0\right)}$

For the following exercises, find the function if $\sin t=\frac{x}{x+1}$.

43. $\cos t$

44. $\sec t$

45. $\cot t$

46. $\cos\left(\sin^{−1}\left(\frac{x}{x+1}\right)\right)$

47. $\tan^{−1}\left({x}{\sqrt{2x+1}}\right)$

48. Graph $y=\sin^{−1}x$ and state the domain and range of the function.

49. Graph $y=\arccos x$ and state the domain and range of the function.

50. Graph one cycle of $y=\tan^{−1}x$ and state the domain and range of the function.

51. For what value of x does $\sin x=\sin^{−1}x$? Use a graphing calculator to approximate the answer.

52. For what value of x does $\cos x=\cos^{−1}x$? Use a graphing calculator to approximate the answer.

53. Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-ﬂoor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

54. Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?

55. An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

56. Without using a calculator, approximate the value of $\arctan(10,000)$. Explain why your answer is reasonable.

57. A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

58. The line $y=\frac{3}{5}x$ passes through the origin in the xy-plane. What is the measure of the angle that the line makes with the positive x-axis?

59. The line $y=−\frac{3}{7}x$ passes through the origin in the xy-plane. What is the measure of the angle that the line makes with the negative x-axis?

60. What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example, a 5% grade means that the road rises 5 feet for every 100 feet of horizontal distance.)

61. A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder’s angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

62. Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?