Solutions for Inverse Trigonometric Functions

Solutions to Try Its

1. [latex]\arccos(0.8776)\approx0.5[/latex]

2. a. [latex]−\frac{\pi}{2}[/latex]; b. [latex]−\frac{\pi}{4}[/latex] c. [latex]\pi[/latex] d. [latex]\frac{\pi}{3}[/latex]

3. 1.9823 or 113.578°

4. [latex]\sin^{−1}(0.6)=36.87^{\circ}=0.6435[/latex] radians

5. [latex]\frac{\pi}{8}\text{; }\frac{2\pi}{9}[/latex]

6. [latex]\frac{3\pi}{4}[/latex]

7. [latex]\frac{12}{13}[/latex]

8. [latex]\frac{4\sqrt{2}}{9}[/latex]

9. [latex]\frac{4x}{\sqrt{16x^{2}+1}}[/latex]

Solutions to Odd-Numbered Exercises

1. The function [latex]y=\sin x[/latex] is one-to-one on [latex]\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right][/latex]; thus, this interval is the range of the inverse function of [latex]y=\sin x\text{, }f\left(x\right)=\sin^{−1}x[/latex]. The function [latex]y=\cos x[/latex] is one-to-one on [0,π]; thus, this interval is the range of the inverse function of [latex]y=\cos x\text{, }f(x)=\cos^{−1}x[/latex].

3. [latex]\frac{\pi}{6}[/latex] is the radian measure of an angle between [latex]−\frac{\pi}{2}[/latex] and [latex]\frac{\pi}{2}[/latex] whose sine is 0.5.

5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [latex]\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right][/latex] so that it is one-to-one and possesses an inverse.

7. True. The angle, [latex]\theta_{1}[/latex] that equals [latex]\arccos(−x)\text{, }x\text{>}0[/latex], will be a second quadrant angle with reference angle, [latex]\theta_{2}[/latex], where [latex]\theta_{2}[/latex] equals [latex]\arccos x\text{, }x\text{>}0[/latex]. Since [latex]\theta_{2}[/latex] is the reference angle for [latex]\theta_{1}[/latex], [latex]\theta_{2}=\pi(−x)=\pi−\arccos x[/latex]

9. [latex]−\frac{\pi}{6}[/latex]

11. [latex]\frac{3\pi}{4}[/latex]

13. [latex]−\frac{\pi}{3}[/latex]

15. [latex]\frac{\pi}{3}[/latex]

17. 1.98

19. 0.93

21. 1.41

23. 0.56 radians

25. 0

27. 0.71

29. −0.71

31. [latex]−\frac{\pi}{4}[/latex]

33. 0.8

35. [latex]\frac{5}{13}[/latex]

37. [latex]\frac{x−1}{\sqrt{−x^{2}+2x}}[/latex]

39. [latex]\frac{\sqrt{x^{2}−1}}{x}[/latex]

41. [latex]\frac{x+0.5}{\sqrt{−x^{2}−x+\frac{3}{4}}}[/latex]

43. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]

45. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]

47. t

49. domain [−1,1]; range [0,π]
A graph of the function arc cosine of x over −1 to 1. The range of the function is 0 to pi.

51. approximately [latex]x=0.00[/latex]

53. 0.395 radians

55. 1.11 radians

57. 1.25 radians

59. 0.405 radians

61. No. The angle the ladder makes with the horizontal is 60 degrees.