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 [latex]y=−1[/latex], becoming an increasing function.

3. [latex]g(x)=4\tan(2x)[/latex]

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

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

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. [latex]−\cot x\cos x−\sin x[/latex]

19. stretching factor: 2; period: [latex]\frac{\pi}{4}[/latex]; asymptotes: [latex]x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8[/latex], 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: [latex]x=3k[/latex], 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: [latex]x=πk[/latex], 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: [latex]x=\frac{\pi}{4}+{\pi}k[/latex], 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: [latex]x=πk[/latex], 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: [latex]\frac{2\pi}{3}[/latex]; asymptotes: [latex]x=\frac{\pi}{6}k[/latex], 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: [latex]\frac{2\pi}{5}[/latex]; asymptotes: [latex]x=\frac{\pi}{10}k[/latex], 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: [latex]x=−\frac{\pi}{4}+\pi k[/latex], where k is an integer
A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.

35. stretching factor: [latex]\frac{7}{5}[/latex]; period: 2π; asymptotes: [latex]x=\frac{\pi}{4}+\pi[/latex]k, where k is an integer
A graph of a modified cosecant function. Four vertical asymptotes.

37. [latex]y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2[/latex]
A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.

39. [latex]f(x)=\csc(2x)[/latex]

41. [latex]f(x)=\csc(4x)[/latex]

43. [latex]f(x)=2\csc x[/latex]

45. [latex]f(x)=\frac{1}{2}\tan(100\pi x)[/latex]

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 [latex]\csc x[/latex] as [latex]\frac{1}{\sin x}[/latex].

46. [latex]f(x)=|\csc(x)|[/latex]

47. [latex]f(x)=|\cot(x)|[/latex]

48. [latex]f(x)=2^{\csc(x)}[/latex]

49. [latex]f(x)=\frac{\csc(x)}{\sec(x)}[/latex]

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. [latex](−\frac{\pi}{2}\text{,}\frac{\pi}{2})[/latex];
b.
A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.
c. [latex]x=−\frac{\pi}{2}[/latex] and [latex]x=\frac{\pi}{2}[/latex]; the distance grows without bound as |x| approaches [latex]\frac{\pi}{2}[/latex]—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 [latex]x=−\frac{\pi}{3}[/latex], the boat is 3 km away;
e. 1.73; when [latex]x=\frac{\pi}{6}[/latex], the boat is about 1.73 km away;
f. 1.5 km; when [latex]x=0[/latex].

57. a. [latex]h(x)=2\tan\left(\frac{\pi}{120}x\right)[/latex];
b.
An exponentially increasing function with a vertical asymptote at x=60.
c. [latex]h(0)=0:[/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[/latex] after 30 seconds, the rockets is 2 mi high;
d. As x approaches 60 seconds, the values of [latex]h(x)[/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.