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Solutions for Inverse Trigonometric Functions

Solutions to Try Its

1. arccos(0.8776)0.5\arccos(0.8776)\approx0.5 2. a. π2−\frac{\pi}{2}; b. π4−\frac{\pi}{4} c. π\pi d. π3\frac{\pi}{3} 3. 1.9823 or 113.578° 4. sin1(0.6)=36.87=0.6435\sin^{−1}(0.6)=36.87^{\circ}=0.6435 radians 5. π82π9\frac{\pi}{8}\text{; }\frac{2\pi}{9} 6. 3π4\frac{3\pi}{4} 7. 1213\frac{12}{13} 8. 429\frac{4\sqrt{2}}{9} 9. 4x16x2+1\frac{4x}{\sqrt{16x^{2}+1}}

Solutions to Odd-Numbered Exercises

1. The function y=sinxy=\sin x is one-to-one on [π2π2]\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right]; thus, this interval is the range of the inverse function of y=sinxf(x)=sin1xy=\sin x\text{, }f\left(x\right)=\sin^{−1}x. The function y=cosxy=\cos x is one-to-one on [0,π]; thus, this interval is the range of the inverse function of y=cosxf(x)=cos1xy=\cos x\text{, }f(x)=\cos^{−1}x. 3. π6\frac{\pi}{6} is the radian measure of an angle between π2−\frac{\pi}{2} and π2\frac{\pi}{2} 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 [π2π2]\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right] so that it is one-to-one and possesses an inverse. 7. True. The angle, θ1\theta_{1} that equals arccos(x)x>0\arccos(−x)\text{, }x\text{>}0, will be a second quadrant angle with reference angle, θ2\theta_{2}, where θ2\theta_{2} equals arccosxx>0\arccos x\text{, }x\text{>}0. Since θ2\theta_{2} is the reference angle for θ1\theta_{1}, θ2=π(x)=πarccosx\theta_{2}=\pi(−x)=\pi−\arccos x 9. π6−\frac{\pi}{6} 11. 3π4\frac{3\pi}{4} 13. π3−\frac{\pi}{3} 15. π3\frac{\pi}{3} 17. 1.98 19. 0.93 21. 1.41 23. 0.56 radians 25. 0 27. 0.71 29. −0.71 31. π4−\frac{\pi}{4} 33. 0.8 35. 513\frac{5}{13} 37. x1x2+2x\frac{x−1}{\sqrt{−x^{2}+2x}} 39. x21x\frac{\sqrt{x^{2}−1}}{x} 41. x+0.5x2x+34\frac{x+0.5}{\sqrt{−x^{2}−x+\frac{3}{4}}} 43. 2x+1x+1\frac{\sqrt{2x+1}}{x+1} 45. 2x+1x+1\frac{\sqrt{2x+1}}{x+1} 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 x=0.00x=0.00 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.

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.