Stewart Precalc 6e Section 5.4: More Trigonometric Graphs

1 Concept - Tangent Period and Asymptotes · Level 1

The trigonometric function \(y = \tan x\) has period ____ and asymptotes \(x =\) ____. Sketch a graph of this function on the interval \(\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\).

2 Concept - Cosecant Period and Asymptotes · Level 1

The trigonometric function \(y = \csc x\) has period ____ and asymptotes \(x =\) ____. Sketch a graph of this function on the interval \((-\pi, \pi)\).

3 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = \tan\left(x + \dfrac{\pi}{4}\right)\).

4 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = \sec(2 x)\).

5 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = \cot(2 x)\).

6 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = -\tan x\).

7 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = 2 \sec x\).

8 Skill - Matching Trig Graphs · Level 2

Match the trigonometric function with one of the graphs I–VI: \(f(x) = 1 + \csc x\).

9 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 4 \tan x\).

10 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = -4 \tan x\).

11 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = -\left(\dfrac{1}{2}\right) \tan x\).

12 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \left(\dfrac{1}{2}\right) \tan x\).

13 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = -\cot x\).

14 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 2 \cot x\).

15 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 2 \csc x\).

16 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \left(\dfrac{1}{2}\right) \csc x\).

17 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 3 \sec x\).

18 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = -3 \sec x\).

19 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \tan\left(x + \dfrac{\pi}{2}\right)\).

20 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \tan\left(x - \dfrac{\pi}{4}\right)\).

21 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \csc\left(x - \dfrac{\pi}{2}\right)\).

22 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \sec\left(x + \dfrac{\pi}{4}\right)\).

23 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \cot\left(x + \dfrac{\pi}{4}\right)\).

24 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 2 \csc\left(x - \dfrac{\pi}{3}\right)\).

25 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = \left(\dfrac{1}{2}\right) \sec\left(x - \dfrac{\pi}{6}\right)\).

26 Skill - Period and Graph · Level 2

Find the period and graph the function \(y = 3 \csc\left(x + \dfrac{\pi}{2}\right)\).

27 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(4 x)\).

28 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(\left(\dfrac{1}{2}\right) x)\).

29 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(\left(\dfrac{\pi}{4}\right) x)\).

30 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \cot(\left(\dfrac{\pi}{2}\right) x)\).

31 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \sec(2 x)\).

32 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 5 \csc(3 x)\).

33 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \csc(4 x)\).

34 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \csc(\left(\dfrac{1}{2}\right) x)\).

35 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 2 \tan(3 \pi x)\).

36 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 2 \tan(\left(\dfrac{\pi}{2}\right) x)\).

37 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 5 \csc(((3 \pi)/2) x)\).

38 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 5 \sec(2 \pi x)\).

39 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(2 \left(x + \dfrac{\pi}{2}\right))\).

40 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \csc(2 \left(x + \dfrac{\pi}{2}\right))\).

41 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(2 (x - \pi))\).

42 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \sec(2 \left(x - \dfrac{\pi}{2}\right))\).

43 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \cot\left(2 x - \dfrac{\pi}{2}\right)\).

44 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \left(\dfrac{1}{2}\right) \tan(\pi x - \pi)\).

45 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 2 \csc\left(\pi x - \dfrac{\pi}{3}\right)\).

46 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 2 \sec(\left(\dfrac{1}{2}\right) x - \dfrac{\pi}{3})\).

47 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 5 \sec\left(3 x - \dfrac{\pi}{2}\right)\).

48 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \left(\dfrac{1}{2}\right) \sec(2 \pi x - \pi)\).

49 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(\left(\dfrac{2}{3}\right) x - \dfrac{\pi}{6})\).

50 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \tan(\left(\dfrac{1}{2}\right)\left(x + \dfrac{\pi}{4}\right))\).

51 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 3 \sec(\pi \left(x + \dfrac{1}{2}\right))\).

52 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = \sec\left(3 x + \dfrac{\pi}{2}\right)\).

53 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = -2 \tan\left(2 x - \dfrac{\pi}{3}\right)\).

54 Skill - Period and Graph · Level 3

Find the period and graph the function \(y = 2 \csc(3 x + 3)\).

55 Skill - Proof of Periodicity · Level 4

(a) Prove that if \(f\) is periodic with period \(p\), then \(\dfrac{1}{f}\) is also periodic with period \(p\). (b) Prove that cosecant and secant each have period \(2 \pi\).

56 Skill - Proof of Periodicity · Level 4

Prove that if \(f\) and \(g\) are periodic with period \(p\), then \(\dfrac{f}{g}\) is also periodic, but its period could be smaller than \(p\).

57 Application - Lighthouse Beam · Level 4

The beam from a lighthouse completes one rotation every two minutes. At time \(t\), the distance \(d\) shown in the figure is \(d(t) = 3 \tan(\pi t)\), where \(t\) is in minutes and \(d\) in miles.

(a) Find \(d(0.15)\), \(d(0.25)\), and \(d(0.45)\).

Answer for 57(a)

(b) Sketch a graph of the function \(d\) for \(0 \leq t < \dfrac{1}{2}\).

Answer for 57(b)

(c) What happens to the distance \(d\) as \(t\) approaches \(\dfrac{1}{2}\)?

Answer for 57(c)

Enter your answer directly below each part above.

58 Application - Length of Shadow · Level 4

On a day when the sun passes directly overhead at noon, a six-foot-tall man casts a shadow of length \(S(t) = 6 |\cot(\left(\dfrac{\pi}{12}\right) t)|\), where \(S\) is in feet and \(t\) is the number of hours since 6 A.M.

(a) Find the length of the shadow at 8:00 A.M., noon, 2:00 P.M., and 5:45 P.M.

Answer for 58(a)

(b) Sketch a graph of \(S\) for \(0 < t < 12\).

Answer for 58(b)

(c) From the graph determine the values of \(t\) at which the length of the shadow equals the man's height. To what time of day does each of these values correspond?

Answer for 58(c)

(d) Explain what happens to the shadow as the time approaches 6 P.M. (that is, as \(t \rightarrow 12^-\)).

Answer for 58(d)

Enter your answer directly below each part above.

59 Discovery - Reduction Formulas · Level 3

Use the graphs in Figure 5 to explain why the following formulas are true: \(\tan\left(x - \dfrac{\pi}{2}\right) = -\cot x\) \(\sec\left(x - \dfrac{\pi}{2}\right) = \csc x\)

60 Example - Graphing Tangent Curves · Level 2

Graph each function. (a) \(y = 2 \tan x\) (b) \(y = -\tan x\)

61 Example - Graphing Tangent Curves with Period Change and Shift · Level 3

Graph each function. (a) \(y = \tan 2x\) (b) \(y = \tan 2\left(x - \dfrac{\pi}{4}\right)\)

62 Example - Graphing Cotangent Curves with Shift · Level 3

Graph \(y = 2 \cot\left(3x - \dfrac{\pi}{2}\right)\).

63 Example - Graphing Cosecant Curves · Level 3

Graph each function. (a) \(y = \dfrac{1}{2} \csc 2x\) (b) \(y = \dfrac{1}{2} \csc\left(2x + \dfrac{\pi}{2}\right)\)

64 Example - Graphing a Secant Curve · Level 3

Graph \(y = 3 \sec(\left(\dfrac{1}{2}\right) x)\).

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