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419
5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
After determining A, B, k, and C, write the resulting
equation.
duce a scatter plot in the viewing window. Choose
0 y 80 for the viewing window.
(C) Plot the results of parts A and B in the same viewing
window. (An improved fit may result by adjusting your
value of C slightly.)
(B) It appears that a sine curve of the form
y k A sin (Bx C)
TABLE 2
will closely model this data. The constants k, A, and B
are easily determined from Table 2 as follows: A (Max y Min y)/2, B 2/Period, k Min y A. To
estimate C, visually estimate to one decimal place the
smallest positive phase shift from the plot in part A.
SECTION
5-8
x (mos.)
1
2
3
4
5
6
7
8
9
10 11 12
y (temp.) 31 34 43 53 62 71 76 74 67 55 45 35
Graphing More General Tangent,
Cotangent, Secant, and Cosecant Functions
• Graphing y A tan (Bx C ) and y A cot (Bx C )
• Graphing y A sec (Bx C ) and y A csc (Bx C )
In this section the graphing of the more general forms of the tangent, cotangent,
secant, and cosecant functions is discussed. Essentially, we follow the same process
we developed for graphing y A sin (Bx C ) and y A cos (Bx C ). The process
is not difficult if you have a clear understanding of the basic graphs and periodic properties for each of these functions.
• Graphing y A tan (Bx C ) and
y A cot (Bx C )
For convenient reference, we repeat the graphs that were shown for y tan x and
y cot x in Section 5-6 (see Figs. 1 and 2).
Graph of y tan x
y
2
5
2
FIGURE 1
1
3
2
2
0
1
2
2
3
2
5
2
x
Period: Domain: All real numbers except
/2 k, k an integer
Range: All real numbers
Symmetric with respect to the origin
Increasing function between
consecutive asymptotes
Discontinuous at x /2 k,
k an integer
420
5 Trigonometric Functions
Graph of y cot x
y
1
2
3
2
2
0
1
2
x
3
2
2
5
2
3
Period: Domain: All real numbers
except k, k an
integer
Range: All real numbers
Symmetric with respect to the
origin
Decreasing function between
consecutive asymptotes
Discontinuous at x k,
k an integer
FIGURE 2
EXPLORE-DISCUSS 1
(A) Match each function to its graph and discuss how the graph compares to the
graph of y tan x or y cot x.
(1) y 4 tan x
(2) y tan 2x
(3) y cot (x /2)
y
y
y
10
10
10
5
5
5
0
x
0
x
5
5
0
x
5
10
10
(a)
(b)
(c)
(B) Use a graphing utility to explore the nature of the changes in the graphs of
the following functions when the values of A, B, and C are changed. Discuss
what happens in each case.
y A tan x and y A cot x for different values of A
y tan Bx and y cot Bx for different values of B
y tan (x C ) and y cot (x C ) for different values of C
To quickly sketch the graphs of equations of the form y A tan (Bx C ), you
need to know how the constants A, B, and C affect the basic graphs of y tan x and
y cot x, respectively.
First note that amplitude is not defined for the tangent and cotangent functions. The graphs of both deviate without end from the x axis. The effect of A is to
5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
421
make the graph steeper if A 1 or to make the graph less steep if A 1. If A is
negative, the graph is reflected across the x axis.
Just as with the sine and cosine functions, the constants B and C, respectively,
effect a change in period and phase shift. Since A tan x and A cot x each has a period
of , it follows that A tan (Bx C ) and A cot (Bx C ) each completes one cycle
as Bx C varies from
Bx C 0
to
Bx C or (solving for x) as x varies from
↓
x
C
B
Phase shift
↓
x
to
Period
↓
C B B
Thus, y A tan (Bx C ) and y A cot (Bx C ) each has a period of /B and
a phase shift of C/B. The basic graph is shifted to the right if C/B is positive and
to the left if C/B is negative.
As before, you do not need to memorize the formulas for period and phase shift.
You only need to remember the process used to obtain the formulas.
EXAMPLE 1
Graphing an Equation of the Form y A cot (Bx C )
Find the period and phase shift for y 2 cot (x/2), then sketch its graph for 2 x 2.
Solution
One cycle of y 2 cot (x/2) is completed as x/2 varies from 0 to . Solve each equation for x:
x
0
2
x
2
x0
x 0 2
Phase shift 0
Period 2
In general, if C 0, there is no phase shift. The graph is sketched for one period,
(0, 2), then extended over the interval (2, 2) as in Figure 3.
FIGURE 3
y
2
0
y
2
x
2
0
2
x
422
5 Trigonometric Functions
Matched Problem 1
EXAMPLE 2
Find the period and phase shift for y 3 tan (x/2), then sketch its graph for 3 x 3.
Graphing an Equation of the Form y A cot (Bx C )
Find the period and phase shift for y cot (2x /2), then sketch the graph for
/2 x .
Solution
Step 1. Find the period and phase shift by solving Bx C 0 and Bx C for x:
2x y
4
4
2
2
2x 2
x
4
x
4
2
Phase shift 4
x
5
FIGURE 4
2x 2x 5
0
2
Period 2
Step 2. Sketch one period of the graph starting at x /4 (the phase shift) and
ending at x /4 /2 (the phase shift plus one period)—Figure 4.
Step 3. Extend the graph over the interval (/2, )—Figure 5.
y
FIGURE 5
5
2
4
4
2
3
4
x
5
Matched Problem 2
Find the period and phase shift for y tan (x/2 /4), then sketch the graph for
3 x 3.
• Graphing y A sec (Bx C ) and
y A csc (Bx C )
For convenient reference, we repeat the graphs that were shown for y csc x and
y sec x in Section 5-6 (see Figs. 6 and 7).
423
5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
Graph of y csc x
y
y csc x Period: 2
Domain: All real numbers except k,
k an integer
Range: All real numbers y such that
y 1 or y 1
Symmetric with respect to the origin
Discontinuous at x k, k an integer
1
sin x
y sin x
2
3
2
2
3
2
1
0
1
2
x
2
FIGURE 6
Graph of y sec x
y
y sec x Period: 2
Domain: All real numbers except
/2 k, k an integer
Range: All real numbers y such that
y 1 or y 1
Symmetric with respect to the y axis
Discontinuous at x /2 k,
k an integer
1
cos x
y cos x
1
2
3
2
2
0
1
2
3
2
x
2
FIGURE 7
EXPLORE-DISCUSS 2
(A) Match each function to its graph, and discuss how the graph compares to the
graph of y csc x or y sec x.
(1) y 21 csc x
(2) y sec x
(3) y csc (x /2)
y
y
5
y
5
3
2
1
1
2
0
x
1
2
3
2
2
0
2
x
3
2
1
2
5
5
(a)
3
(b)
(c)
x
2
424
5 Trigonometric Functions
(B) Use a graphing utility to explore the nature of the changes in the graphs of
the following functions when the values of A, B, and C are changed. Discuss
what happens in each case.
y A sec x and y A csc x for different values of A
y sec Bx and y csc Bx for different values of B
y sec (x C ) and y csc (x C ) for different values of C
As with the tangent and cotangent functions, amplitude is not defined for either
the secant or the cosecant functions. Since both functions have a period of 2, we
find the period and phase shift for each by solving Bx C 0 and Bx C 2.
To graph either y A sec (Bx C ) or y A csc (Bx C ), you will probably
find it easier to graph y (1/A) cos (Bx C ) or y (1/A) sin (Bx C ) with a
dashed curve, then take reciprocals. An example should help to make the process clear.
EXAMPLE 3
Graphing an Equation of the Form y A sec (Bx C )
Find the period and phase shift for y 3/4 x 3/4.
Solution
1
2
sec (2x ), then sketch the graph for
Step 1. Find the period and phase shift by solving Bx C 0 and Bx C 2
for x:
2x 0
2x 2
2x 2x 2
x
2
x
Phase shift 2
Period 2
Step 2. Since
1
1
sec (2x ) 2
2 cos (2x )
we graph
y 2 cos (2x )
for one cycle from /2 to /2 , and then take reciprocals. Notice
that we also place vertical asymptotes through the x intercepts of the cosine
graph to guide us when we sketch the secant function—Figure 8.
425
5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
FIGURE 8
y
4
3
2
1
2
2
1
x
3
4
Step 3. Extend the graph over the required interval (3/4, 3/4)—Figure 9.
y
FIGURE 9
4
3
2
1
2
x
2
1
3
4
Matched Problem 3
Find the period and phase shift for y 2 csc (x/2 ), then sketch the graph for
2 x 10.
Answers to Matched Problems
2. Period 2, phase shift 21
1. Period 2, phase shift 0
y
y
6
3
2
6
x
1
1
6
2
3
1
3
1
2
x
2
6
3
426
5 Trigonometric Functions
3. Period 4, phase shift 2
y
1
2
EXERCISE
x
1
2
4
6
8
10
5-8
A
In Problems 15–18, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
In Problems 1–8, find the period of each function, and
graph the function for the indicated interval.
15. The graphs of y cos (x) and y csc (x) have infinitely
many intersection points.
1. y 2 cot 4x, 0 x /2
2. y 3 tan 2x, x 3. y 14 tan 8x, 0 x 21
4. y 12
cot 2x, 0 x 1
5. y csc (x/2), 3 x 3
6. y sec x, 1.5 x 3.5
7. y 2 sec x, 1 x 3
8. y 2 csc (x/2), 0 x 8
B
In Problems 9–14, find the period and phase shift, then
graph each function.
2 , 2 x 32
10. y tan x , x 2
16. The graphs of y sin (x) and y csc (x) have infinitely
many intersection points.
17. Every horizontal line intersects the graph of y 0.1 sec (5x 1) infinitely many times.
18. The maximum deviation of the graph of y 7 tan (3x 2) from the x axis is 7.
In Problems 19–22, graph at least two cycles of the given
equation in a graphing utility, then find an equation of the
form y A tan Bx, y A cot Bx, y A sec Bx, or y A csc Bx that has the same graph. (These problems suggest
additional identities beyond those discussed in Section 5-2.
Additional identities are discussed in detail in Chapter 6.)
19. y cot x tan x
20. y cot x tan x
21. y csc x cot x
22. y csc x cot x
9. y cot x 3
3
x
11. y tan (2x ), 4
4
12. y cot (2x ), x
2
2
14. y csc x , 1 x 1
2
13. y sec x , 1 x 1
2
C
In Problems 23–26, find the period and phase shift, then
graph each function.
23. y 2 tan
4 x 4 , 1 x 7
24. y 3 cot (x ), 2 x 2
2 x 2 , 1 x 3
26. y 2 sec x , 1 x 3
2
25. y 3 csc
5-9 Inverse Trigonometric Functions
(B) Graph the equation found in part A for the time interval
[0, 1). If the graph has an asymptote, put it in.
In Problems 27–30, graph at least two cycles of the given
equation in a graphing utility, then find an equation of the
form y A tan Bx, y A cot Bx, y A sec Bx, or y A csc Bx that has the same graph. (These problems suggest
additional identities beyond those discussed in Section 5-2.
Additional identities are discussed in detail in Chapter 6.)
(C) Describe what happens to the length c of the light
beam as t goes from 0 to 1.
27. y sin 3x cos 3x cot 3x
P
28. y cos 2x sin 2x tan 2x
c
20
sin 6x
1 cos 6x
★
APPLICATIONS
★
a
N
sin 4x
29. y 1 cos 4x
30. y 427
31. Motion. A beacon light 20 ft from a wall rotates clockwise
at the rate of 1/4 rps (see figure); thus, t/2.
(A) Start counting time in seconds when the light spot is at
N and write an equation for the length c of the light
beam in terms of t.
SECTION
5-9
32. Motion. Refer to Problem 31.
(A) Write an equation for the distance a the light spot travels along the wall in terms of time t.
(B) Graph the equation found in part A for the time interval
[0, 1). If the graph has an asymptote, put it in.
(C) Describe what happens to the distance a along the wall
as t goes from 0 to 1.
Inverse Trigonometric Functions
•
•
•
•
•
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
Summary
Inverse Cotangent, Secant, and Cosecant Functions (Optional)
A brief review of the general concept of inverse functions discussed in Section 2-6
should prove helpful before proceeding with this section. In the following box we
restate a few important facts about inverse functions from that section.
Facts about Inverse Functions
For f a one-to-one function and f 1 its inverse:
1. If (a, b) is an element of f, then (b, a) is an element of f 1, and conversely.
2. Range of f Domain of f 1
Domain of f Range of f 1