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SECTION 4.5
Graphs of Tangent, Cotangent, Secant, and Cosecant
361
4.5 Graphs of Tangent, Cotangent,
Secant, and Cosecant
What you’ll learn about
• The Tangent Function
• The Cotangent Function
• The Secant Function
The Tangent Function
The graph of the tangent function is shown below. As with the sine and cosine graphs,
this graph tells us quite a bit about the function’s properties. Here is a summary of
tangent facts:
• The Cosecant Function
... and why
This will give us functions for the
remaining trigonometric ratios.
THE TANGENT FUNCTION
[–3π /2, 3π /2] by [–4, 4]
FIGURE 4.44A
ƒ1x2 = tan x
Domain: All reals except odd multiples of p/2
Range: All reals
Continuous (i.e., continuous on its domain)
Increasing on each interval in its domain
Symmetric with respect to the origin (odd)
Not bounded above or below
No local extrema
No horizontal asymptotes
Vertical asymptotes: x = k # 1p/22 for all odd integers k
End behavior: lim tan x and lim tan x do not exist. (The function values
x: -q
x: q
continually oscillate between - q and q and approach no limit.)
y
3
2
–2π
π
x
tan x =
–3
FIGURE 4.45 The tangent function has
asymptotes at the zeros of cosine.
y
3
2
1
–2π
We now analyze why the graph of ƒ1x2 = tan x behaves the way it does. It follows
from the definitions of the trigonometric functions (Section 4.2) that
π
x
–3
FIGURE 4.46 The tangent function has
zeros at the zeros of sine.
sin x
.
cos x
Unlike the sinusoids, the tangent function has a denominator that might be zero, which
makes the function undefined. Not only does this actually happen, but it happens an infinite number of times: at all the values of x for which cos x = 0. That is why the tangent function has vertical asymptotes at those values (Figure 4.45). The zeros of the
tangent function are the same as the zeros of the sine function: all the integer multiples
of p (Figure 4.46).
Because sin x and cos x are both periodic with period 2p, you might expect the period
of the tangent function to be 2p also. The graph shows, however, that it is p.
The constants a, b, h, and k influence the behavior of y = a tan 1b1x - h22 + k in
much the same way that they do for the graph of y = a sin 1b1x - h22 + k. The constant a yields a vertical stretch or shrink, b affects the period, h causes a horizontal
translation, and k causes a vertical translation. The terms amplitude and phase shift,
however, are not used, as they apply only to sinusoids.
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CHAPTER 4 Trigonometric Functions
EXAMPLE 1 Graphing a Tangent Function
Describe the graph of the function y = - tan 2x in terms of a basic trigonometric
function. Locate the vertical asymptotes and graph four periods of the function.
SOLUTION The effect of the 2 is a horizontal shrink of the graph of y = tan x by a
factor of 1/2, while the effect of the -1 is a reflection across the x-axis. Since the vertical asymptotes of y = tan x are all odd multiples of p/2, the shrink factor causes the
vertical asymptotes of y = tan 2x to be all odd multiples of p/4 (Figure 4.47a). The
reflection across the x-axis (Figure 4.47b) does not change the asymptotes.
Since the period of the function y = tan x is p, the period of the function y = - tan 2x
is (thanks again to the shrink factor) p/2. Thus, any window of horizontal length 2p
will show four periods of the graph. Figure 4.47b uses the window 3- p, p4 by
3-4, 44.
Now try Exercise 5.
[– π , π ] by [–4, 4]
(a)
The other three trigonometric functions (cotangent, secant, and cosecant) are reciprocals of tangent, cosine, and sine, respectively. (This is the reason that you probably do
not have buttons for them on your calculators.) As functions they are certainly interesting, but as basic functions they are unnecessary—we can do our trigonometric modeling and equation solving with the other three. Nonetheless, we give each of them a brief
section of its own in this book.
The Cotangent Function
The cotangent function is the reciprocal of the tangent function. Thus,
cot x =
[– π , π ] by [–4, 4]
(b)
cos x
.
sin x
The graph of y = cot x will have asymptotes at the zeros of the sine function (Figure 4.48)
and zeros at the zeros of the cosine function (Figure 4.49).
FIGURE 4.47 The graph of (a) y = tan 2x
is reflected across the x-axis to produce the
graph of (b) y = - tan 2x. (Example 1)
y
y
3
2
1
3
2
x
–2π
–3
FIGURE 4.48 The cotangent has
asymptotes at the zeros of the sine function.
Cotangent on the Calculator
If your calculator does not have a “cotan” button,
you can use the fact that cotangent and tangent are
reciprocals. For example, the function in Example 2 can be entered in a calculator as y =
3/tan 1x/22 + 1 or as y = 31tan1x/222-1 + 1.
Remember that it cannot be entered as y =
3 tan-1 1x/22 + 1. (The - 1 exponent in that position represents a function inverse, not a reciprocal.)
–2π
2π
x
–3
FIGURE 4.49 The cotangent has
zeros at the zeros of the cosine function.
EXAMPLE 2 Graphing a Cotangent Function
Describe the graph of ƒ1x2 = 3 cot 1x/22 + 1 in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods.
SOLUTION The graph is obtained from the graph of y = cot x by effecting a hori-
zontal stretch by a factor of 2, a vertical stretch by a factor of 3, and a vertical translation up 1 unit. The horizontal stretch makes the period of the function 2p (twice
the period of y = cot x), and the asymptotes are at the even multiples of p. Figure
4.50 shows two periods of the graph of ƒ.
Now try Exercise 9.
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SECTION 4.5
Graphs of Tangent, Cotangent, Secant, and Cosecant
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The Secant Function
Important characteristics of the secant function can be inferred from the fact that it is
the reciprocal of the cosine function.
Whenever cos x = 1, its reciprocal sec x is also 1. The graph of the secant function has
asymptotes at the zeros of the cosine function. The period of the secant function is 2p,
the same as its reciprocal, the cosine function.
The graph of y = sec x is shown with the graph of y = cos x in Figure 4.51. A local
maximum of y = cos x corresponds to a local minimum of y = sec x, while a local
minimum of y = cos x corresponds to a local maximum of y = sec x.
[–2π , 2π ] by [–10, 10]
FIGURE 4.50 Two periods of
ƒ1x2 = 3 cot 1x/22 + 1. (Example 2)
y
–1
–2
–3
Proving a Graphical Hunch
Figure 4.52 shows that the graphs of y = sec x and y = - 2 cos x never seem to
intersect.
3
2
–2π
EXPLORATION 1
π
2π
x
If we stretch the reflected cosine graph vertically by a large enough number,
will it continue to miss the secant graph? Or is there a large enough (positive)
value of k so that the graph of y = sec x does intersect the graph of
y = - k cos x?
1. Try a few other values of k in your calculator. Do the graphs intersect?
FIGURE 4.51 Characteristics of the
secant function are inferred from the fact that
it is the reciprocal of the cosine function.
2. Your exploration should lead you to conjecture that the graphs of y = sec x
and y = - k cos x will never intersect for any positive value of k. Verify this
conjecture by proving algebraically that the equation
- k cos x = sec x
has no real solutions when k is a positive number.
EXAMPLE 3 Solving a Trigonometric Equation Algebraically
Find the value of x between p and 3p/2 that solves sec x = - 2.
SOLUTION We construct a reference triangle in the third quadrant that has the ap-
propriate ratio, hyp/adj, equal to -2. This is most easily accomplished by choosing
an x-coordinate of -1 and a hypotenuse of 2 (Figure 4.53a). We recognize this as a
30°–60°–90° triangle that determines an angle of 240°, which converts to 4p/3 radians (Figure 4.53b).
Therefore the answer is 4p/3.
y
y
240°
–1
x
x
[–6.5, 6.5] by [–3, 3]
2
FIGURE 4.52 The graphs of y = sec x
and y = - 2 cos x. (Exploration 1)
(a)
(b)
FIGURE 4.53 A reference triangle in the third quadrant (a) with hyp/adj = - 2 determines an angle (b) of 240 degrees, which converts to 4p/3 radians. (Example 3)
Now try Exercise 29.
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CHAPTER 4 Trigonometric Functions
The Cosecant Function
Important characteristics of the cosecant function can be inferred from the fact that it is
the reciprocal of the sine function.
Whenever sin x = 1, its reciprocal csc x is also 1. The graph of the cosecant function
has asymptotes at the zeros of the sine function. The period of the cosecant function is
2p, the same as its reciprocal, the sine function.
The graph of y = csc x is shown with the graph of y = sin x in Figure 4.54. A local
maximum of y = sin x corresponds to a local minimum of y = csc x, while a local
minimum of y = sin x corresponds to a local maximum of y = csc x.
y
3
2
1
x
–2π
–3
FIGURE 4.54 Characteristics of the cosecant function are inferred from the fact that it is the
reciprocal of the sine function.
EXAMPLE 4 Solving a Trigonometric Equation Graphically
Are Cosecant Curves Parabolas?
Find the smallest positive number x such that x 2 = csc x.
Figure 4.55 shows a parabola intersecting
one of the infinite number of U-shaped
curves that make up the graph of the cosecant
function. In fact, the parabola intersects all
of those curves that lie above the x-axis,
since the parabola must spread out to cover
the entire domain of y = x 2, which is all
real numbers! The cosecant curves do not
keep spreading out, as they are hemmed in
by asymptotes. That means that the
U-shaped curves in the cosecant function
are not parabolas.
SOLUTION There is no algebraic attack that looks hopeful, so we solve this equa-
tion graphically. The intersection point of the graphs of y = x 2 and y = csc x that
has the smallest positive x-coordinate is shown in Figure 4.55. We use the grapher to
determine that x L 1.068.
[–6.5, 6.5] by [–3, 3]
FIGURE 4.55 A graphical solution of a trigonometric equation. (Example 3)
Now try Exercise 39.
To close this section, we summarize the properties of the six basic trigonometric functions in tabular form. The “n” that appears in several places should be understood as
taking on all possible integer values: 0, ⫾ 1, ⫾2, ⫾3, Á .
Summary: Basic Trigonometric Functions
Function
Period
Domain
sin x
cos x
tan x
cot x
sec x
csc x
2p
2p
p
p
2p
2p
All reals
All reals
x Z p/2 + np
x Z np
x Z p/2 + np
x Z np
Range
3 - 1, 14
3 - 1, 14
All reals
All reals
1 - q , -1] ´ 31, q 2
1- q , -1] ´ 31, q 2
Asymptotes
None
None
x = p/2 + np
x = np
x = p/2 + np
x = np
Zeros
np
p/2 + np
np
p/2 + np
None
None
Even/Odd
Odd
Even
Odd
Odd
Even
Odd
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SECTION 4.5
QUICK REVIEW 4.5
Graphs of Tangent, Cotangent, Secant, and Cosecant
365
(For help, go to Sections 1.2, 2.6, and 4.3.)
Exercise numbers with a gray background indicate problems
that the authors have designed to be solved without a calculator.
In Exercises 1–4, state the period of the function.
1. y = cos 2x
2. y = sin 3x
1
1
3. y = sin x
4. y = cos x
3
2
In Exercises 5–8, find the zeros and vertical asymptotes of
the function.
x + 5
x - 3
5. y =
6. y =
x + 4
x - 1
7. y =
x + 1
1x - 221x + 22
8. y =
x + 2
x1x - 32
In Exercises 9 and 10, tell whether the function is odd, even, or
neither.
1
9. y = x 2 + 4
10. y =
x
SECTION 4.5 EXERCISES
In Exercises 1–4, identify the graph of each function. Use your understanding of transformations, not your graphing calculator.
3. Graphs of csc x and 3 csc 2x are shown.
y
1. Graphs of one period of csc x and 2 csc x are shown.
10
8
6
4
2
y
10
8
6
4
2
y1
–π
y2
–π
y2
π
y1
π
x
x
–10
4. Graphs of cot x and cot 1x - 0.52 + 3 are shown.
y
2. Graphs of two periods of 0.5 tan x and 5 tan x are shown.
10
8
6
4
y
10
8
6
4
2
–π
y1
–π
π
y2
x
y2
π
–4
–6
–8
–10
y1
–8
–10
x
In Exercises 5–12, describe the graph of the function in terms of a basic
trigonometric function. Locate the vertical asymptotes and graph two
periods of the function.
5. y = tan 2x
6. y = - cot 3x
7. y = sec 3x
8. y = csc 2x
9. y = 2 cot 2x
11. y = csc 1x/22
10. y = 3 tan 1x/22
12. y = 3 sec 4x
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CHAPTER 4 Trigonometric Functions
In Exercises 13–16, match the trigonometric function with its graph.
Then give the Xmin and Xmax values for the viewing window in which
the graph is shown. Use your understanding of transformations, not
your graphing calculator.
39. csc x = 2,
0 … x … 2p
40. tan x = 0.3,
0 … x … 2p
41. Writing to Learn The figure shows a unit circle and
an angle t whose terminal side is in Quadrant III.
y
x2 + y2 = 1
P1(–a, –b)
[?, ?] by [–10, 10]
(b)
[?, ?] by [–10, 10]
(a)
t
t– π
x
P2(a, b)
[?, ?] by [–10, 10]
(d)
[?, ?] by [–10, 10]
(c)
13. y = - 2 tan x
14. y = cot x
15. y = sec 2x
16. y = - csc x
In Exercises 17–20, analyze each function for domain, range, continuity,
increasing or decreasing behavior, symmetry, boundedness, extrema,
asymptotes, and end behavior.
17. ƒ1x2 = cot x
18. ƒ1x2 = sec x
20. ƒ1x2 = tan 1x/22
19. ƒ1x2 = csc x
In Exercises 21–28, describe the transformations required to obtain the
graph of the given function from a basic trigonometric graph.
21. y = 3 tan x
22. y = - tan x
23. y = 3 csc x
24. y = 2 tan x
1
25. y = - 3 cot x
2
p
27. y = - tan x + 2
2
26. y = - 2 sec
(a) If the coordinates of point P2 are 1a, b2, explain why the
coordinates of point P1 on the circle and the terminal side
of angle t - p are 1- a, - b2.
b
(b) Explain why tan t = .
a
(c) Find tan 1t - p2, and show that tan t = tan1t - p2.
(d) Explain why the period of the tangent function is p.
(e) Explain why the period of the cotangent function is p.
42. Writing to Learn Explain why it is correct to say
y = tan x is the slope of the terminal side of angle x in standard position. P is on the unit circle.
y
1
x
2
P(cos x, sin x)
x
28. y = 2 tan px - 2
x
x
In Exercises 29–34, solve for x in the given interval. You should be able
to find these numbers without a calculator, using reference triangles in
the proper quadrants.
29. sec x = 2,
0 … x … p/2
30. csc x = 2,
p/2 … x … p
31. cot x = - 13,
p/2 … x … p
32. sec x = - 12,
p … x … 3p/2
43. Periodic Functions Let ƒ be a periodic function with
period p. That is, p is the smallest positive number such that
33. csc x = 1,
2p … x … 5p/2
ƒ1x + p2 = ƒ1x2
34. cot x = 1,
- p … x … - p/2
In Exercises 35–40, use a calculator to solve for x in the given interval.
p
0 … x …
35. tan x = 1.3,
2
p
0 … x …
36. sec x = 2.4,
2
3p
… x … 2p
37. cot x = - 0.6,
2
3p
38. csc x = - 1.5, p … x …
2
for any value of x in the domain of ƒ. Show that the reciprocal
1/ƒ is periodic with period p.
44. Identities Use the unit circle to give a convincing argument for the identity.
(a) sin 1t + p2 = - sin t
(b) cos 1t + p2 = - cos t
(c) Use (a) and (b) to show that tan 1t + p2 = tan t. Explain
why this is not enough to conclude that the period of
tangent is p.
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SECTION 4.5
45. Lighthouse Coverage The Bolivar Lighthouse is
located on a small island 350 ft from the shore of the mainland
as shown in the figure.
(a) Express the distance d as a function of the angle x.
(b) If x is 1.55 rad, what is d ?
Graphs of Tangent, Cotangent, Secant, and Cosecant
54. Multiple Choice
sects the graph of y =
(A) x.
(B) x 2.
The graph of y = sec x never inter(C) csc x.
(D) cos x. (E) sin x.
55. Multiple Choice
tion y = k csc x?
If k Z 0, what is the range of the func-
(A) 3-k, k4
(B) 1- k, k2
(C) 1- q , -k2 ´ 1k, q 2
350 ft
367
(D) 1- q , -k4 ´ 3k, q 2
(E) 1- q , - 1/k4 ´ 31/k, q 2
x
d
56. Multiple Choice The graph of y = csc x has the same
set of asymptotes as the graph of y =
46. Hot-Air Balloon A hot-air balloon over Albuquerque,
New Mexico, is being blown due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the
ground and the line of vision from P to the balloon. This angle
changes as the balloon travels.
(a) Express the horizontal distance x as a function of the angle
y.
(b) When the angle is p/20 rad, what is its horizontal distance
from P?
(c) An angle of p/20 rad is equivalent to how many degrees?
Wind
blowing
due east
(A) sin x.
(B) tan x.
(D) sec x.
(E) csc 2x.
Explorations
In Exercises 57 and 58, graph both ƒ and g in the 3-p, p4 by 3- 10, 104
viewing window. Estimate values in the interval 3-p, p4 for which
ƒ 7 g.
57. ƒ1x2 = 5 sin x and g1x2 = cot x
58. ƒ1x2 = - tan x and g1x2 = csc x
59. Writing to Learn Graph the function ƒ1x2 = - cot x on
the interval 1-p, p2. Explain why it is correct to say that ƒ is
increasing on the interval 10, p2, but it is not correct to say that
ƒ is increasing on the interval 1 -p, p2.
60. Writing to Learn Graph functions ƒ1x2 = - sec x and
g1x2 =
800 ft
(C) cot x.
1
x - 1p/22
simultaneously in the viewing window 30, p4 by 3- 10, 104.
Discuss whether you think functions ƒ and g are equivalent.
61. Write csc x as a horizontal translation of sec x.
P
y
62. Write cot x as the reflection about the x-axis of a horizontal
translation of tan x.
x
In Exercises 47–50, find approximate solutions for the equation in the
interval - p 6 x 6 p.
47. tan x = csc x
48. sec x = cot x
49. sec x = 5 cos x
50. 4 cos x = tan x
Standardized Test Questions
Extending the Ideas
63. Group Activity Television Coverage A television camera is on a platform 30 m from the point on High
Street where the Worthington Memorial Day Parade will pass.
Express the distance d from the camera to a particular parade
float as a function of the angle x, and graph the function over
the interval -p/2 6 x 6 p/2.
High Street
51. True or False The function ƒ1x2 = tan x is increasing
on the interval 1- q , q 2. Justify your answer.
52. True or False If x = a is an asymptote of the secant
function, then cot a = 0. Justify your answer.
You should answer these questions without using a calculator.
53. Multiple Choice The graph of y = cot x can be obtained by a horizontal shift of the graph of y =
(A) - tan x. (B) - cot x. (C) sec x.
(D) tan x.
(E) csc x.
Float
d
Camera
x
30 m
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CHAPTER 4 Trigonometric Functions
64. What’s in a Name? The word sine comes from the
Latin word sinus, which means “bay” or “cove.” It entered the
language through a mistake (variously attributed to Gerardo of
Cremona or Robert of Chester) in translating the Arabic word
“jiba” (chord) as if it were “jaib” (bay). This was due to the
fact that the Arabs abbreviated their technical terms, much as
we do today. Imagine someone unfamiliar with the technical
term “cosecant” trying to reconstruct the English word that is
abbreviated by “csc.” It might well enter their language as their
word for “cascade.”
The names for the other trigonometric functions can all be
explained.
(a) Cosine means “sine of the complement.” Explain why this
is a logical name for cosine.
(b) In the figure below, BC is perpendicular to OC, which is a
radius of the unit circle. By a familiar geometry theorem,
BC is tangent to the circle. OB is part of a secant that intersects the unit circle at A. It lies along the terminal side of
an angle of t radians in standard position. Write the coordinates of A as functions of t.
where h is the height of the liquid in the tube, r (rho) is the
density of the liquid, g = 9.8 m/sec2 is the acceleration due to
gravity, r is the radius of the tube, and f (phi) is the angle of
contact between the tube and the liquid’s surface. Whole blood
has a surface tension of 0.058 N/m (newton per meter) and a
density of 1050 kg/m3. Suppose that blood rises to a height of
1.5 m in a capillary blood vessel of radius 4.7 * 10 - 6 m. What
is the contact angle between the capillary vessel and the blood
surface? 31N = 11kg # m2/sec24
66. Advanced Curve Fitting A researcher has reason to
believe that the data in the table below can best be described by
an algebraic model involving the secant function:
y = a sec1bx2
Unfortunately, her calculator will do only sine regression. She
realizes that the following two facts will help her:
1
1
1
=
= cos 1bx2
y
a
a sec 1bx2
and
cos 1bx2 = sin abx +
(c) Use similar triangles to find length BC as a trig function of t.
(d) Use similar triangles to find length OB as a trig function of t.
(e) Use the results from parts (a), (c), and (d) to explain where
the names “tangent, cotangent, secant,” and “cosecant”
came from.
y
B
D
C
x
65. Capillary Action A film of liquid
in a thin (capillary) tube has surface tension
g (gamma) given by
g =
1
hrgr sec f,
2
1
1
p
= sin abx + b.
y
a
2
(b) Store the x-values in the table in L1 in your calculator and
the y-values in L2. Store the reciprocals of the y-values in
L3. Then do a sine regression for L3 11/y2 as a function of
L1 1x2. Write the regression equation.
(d) Write the secant model: y = a sec 1bx2 Does the curve fit
the 1L1, L22 scatter plot?
1
t
(a) Use these two facts to show that
(c) Use the regression equation in (b) to determine the values
of a and b.
A
O
p
b
2
x
1
2
3
4
y
5.0703
5.2912
5.6975
6.3622
x
5
6
7
8
y
7.4359
9.2541
12.716
21.255