Solutions to Odd-Numbered Exercises

1.  Since $y=\csc x$ is the reciprocal function of $y=\sin x$, you can plot the reciprocal of the coordinates on the graph of $y=\sin x$ to obtain the y-coordinates of $y=\csc x$. The x-intercepts of the graph $y=\sin x$ are the vertical asymptotes for the graph of $y=\csc x$.

3. Answers will vary. Using the unit circle, one can show that $\tan(x+\pi)=\tan x$.

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. $−\cot x\cos x−\sin x$

19. stretching factor: 2; period: $\frac{\pi}{4}$; asymptotes: $x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8$, where k is an integer

21. stretching factor: 6; period: 6; asymptotes: $x=3k$, where k is an integer

23. stretching factor: 1; period: π; asymptotes: $x=πk$, where k is an integer

25. Stretching factor: 1; period: π; asymptotes: $x=\frac{\pi}{4}+{\pi}k$, where k is an integer

27. stretching factor: 2; period: 2π; asymptotes: $x=πk$, where k is an integer

29. stretching factor: 4; period: $\frac{2\pi}{3}$; asymptotes: $x=\frac{\pi}{6}k$, where k is an odd integer

31. stretching factor: 7; period: $\frac{2\pi}{5}$; asymptotes: $x=\frac{\pi}{10}k$, where k is an odd integer

33. stretching factor: 2; period: 2π; asymptotes: $x=−\frac{\pi}{4}+\pi k$, where k is an integer

35. stretching factor: $\frac{7}{5}$; period: 2π; asymptotes: $x=\frac{\pi}{4}+\pi$k, where k is an integer

37. $y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2$

39. $f(x)=\csc(2x)$

41. $f(x)=\csc(4x)$

43. $f(x)=2\csc x$

45. $f(x)=\frac{1}{2}\tan(100\pi x)$

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 $\csc x$ as $\frac{1}{\sin x}$.

46. $f(x)=|\csc(x)|$

47. $f(x)=|\cot(x)|$

48. $f(x)=2^{\csc(x)}$

49. $f(x)=\frac{\csc(x)}{\sec(x)}$

51.

53.

55. a. $(−\frac{\pi}{2}\text{,}\frac{\pi}{2})$;
b.

c. $x=−\frac{\pi}{2}$ and $x=\frac{\pi}{2}$; the distance grows without bound as |x| approaches $\frac{\pi}{2}$—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 $x=−\frac{\pi}{3}$, the boat is 3 km away;
e. 1.73; when $x=\frac{\pi}{6}$, the boat is about 1.73 km away;
f. 1.5 km; when $x=0$.

57. a. $h(x)=2\tan\left(\frac{\pi}{120}x\right)$;
b.

c. $h(0)=0:$ after 0 seconds, the rocket is 0 mi above the ground; $h(30)=2:$ after 30 seconds, the rockets is 2 mi high;
d. As x approaches 60 seconds, the values of $h(x)$ grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.