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Study Guides > Precalculus II

Solutions for The Other Trigonometric Functions

Solutions to Try Its

1. A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3. 2. It would be reflected across the line y=−1y=−1, becoming an increasing function. 3. g(x)=4tan⁡(2x)g(x)=4\tan(2x) 4. This is a vertical reflection of the preceding graph because A is negative. A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola. 5. A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16. 6. A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola. 7. A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.

Solutions to Odd-Numbered Exercises

1.  Since y=csc⁡xy=\csc x is the reciprocal function of y=sin⁡xy=\sin x, you can plot the reciprocal of the coordinates on the graph of y=sin⁡xy=\sin x to obtain the y-coordinates of y=csc⁡xy=\csc x. The x-intercepts of the graph y=sin⁡xy=\sin x are the vertical asymptotes for the graph of y=csc⁡xy=\csc x. 3. Answers will vary. Using the unit circle, one can show that tan⁡(x+π)=tan⁡x\tan(x+\pi)=\tan x. 5. The period is the same: 2π. 7. IV 9. III 11. period: 8; horizontal shift: 1 unit to left 13. 1.5 15. 5 17. −cot⁡xcos⁡x−sin⁡x−\cot x\cos x−\sin x 19. stretching factor: 2; period: π4\frac{\pi}{4}; asymptotes: x=14(π2+πk)+8x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8, where k is an integer A graph of two periods of a modified tangent function. There are two vertical asymptotes. 21. stretching factor: 6; period: 6; asymptotes: x=3kx=3k, where k is an integer A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6. 23. stretching factor: 1; period: π; asymptotes: x=πkx=πk, where k is an integer A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi. 25. Stretching factor: 1; period: π; asymptotes: x=π4+πkx=\frac{\pi}{4}+{\pi}k, where k is an integer A graph of two periods of a modified tangent function. Three vertical asymptiotes shown. 27. stretching factor: 2; period: 2π; asymptotes: x=πkx=πk, where k is an integer A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi. 29. stretching factor: 4; period: 2π3\frac{2\pi}{3}; asymptotes: x=π6kx=\frac{\pi}{6}k, where k is an odd integer A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2. 31. stretching factor: 7; period: 2π5\frac{2\pi}{5}; asymptotes: x=π10kx=\frac{\pi}{10}k, where k is an odd integer A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart. 33. stretching factor: 2; period: 2π; asymptotes: x=−π4+πkx=−\frac{\pi}{4}+\pi k, where k is an integer A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart. 35. stretching factor: 75\frac{7}{5}; period: 2π; asymptotes: x=π4+πx=\frac{\pi}{4}+\pik, where k is an integer A graph of a modified cosecant function. Four vertical asymptotes. 37. y=tan⁡(3(x−π4))+2y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2 A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12. 39. f(x)=csc⁡(2x)f(x)=\csc(2x) 41. f(x)=csc⁡(4x)f(x)=\csc(4x) 43. f(x)=2csc⁡xf(x)=2\csc x 45. f(x)=12tan⁡(100πx)f(x)=\frac{1}{2}\tan(100\pi x) For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input csc⁡x\csc x as 1sin⁡x\frac{1}{\sin x}. 46. f(x)=∣csc⁡(x)∣f(x)=|\csc(x)| 47. f(x)=∣cot⁡(x)∣f(x)=|\cot(x)| 48. f(x)=2csc⁡(x)f(x)=2^{\csc(x)} 49. f(x)=csc⁡(x)sec⁡(x)f(x)=\frac{\csc(x)}{\sec(x)} 51. A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi. 53. A graph of y=1. 55. a. (−π2,π2)(−\frac{\pi}{2}\text{,}\frac{\pi}{2}); b. A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2. c. x=−π2x=−\frac{\pi}{2} and x=π2x=\frac{\pi}{2}; the distance grows without bound as |x| approaches π2\frac{\pi}{2}—i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it; d. 3; when x=−π3x=−\frac{\pi}{3}, the boat is 3 km away; e. 1.73; when x=π6x=\frac{\pi}{6}, the boat is about 1.73 km away; f. 1.5 km; when x=0x=0. 57. a. h(x)=2tan⁡(π120x)h(x)=2\tan\left(\frac{\pi}{120}x\right); b. An exponentially increasing function with a vertical asymptote at x=60. c. h(0)=0:h(0)=0: after 0 seconds, the rocket is 0 mi above the ground; h(30)=2:h(30)=2: after 30 seconds, the rockets is 2 mi high; d. As x approaches 60 seconds, the values of h(x)h(x) grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

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