Download SECTION 5-6 Graphing Basic Trigonometric Functions

5-6
SECTION
5-6
Graphing Basic Trigonometric Functions
391
Graphing Basic Trigonometric Functions
•
•
•
•
•
Periodic Functions
Graphs of y sin x and y cos x
Graphs of y tan x and y cot x
Graphs of y csc x and y sec x
Graphs on a Graphing Utility
Consider the graphs of sunrise times and sound waves shown in Figure 1. What is a
common feature of the two graphs? Both represent repetitive phenomena; that is, both
appear to be periodic. Trigonometric functions are particularly suited to describe periodic phenomena.
FIGURE 1 Periodic phenomena.
Sunrise time
8
7
6
Pressure at eardrum
5
F
A
J
A
O
D
F
A
J
A
O
D
Month
Sunrise times and time of year
( )
(a)
1
600
1
300
Time (seconds)
Soundwave arriving at eardrum
(b)
This section discusses the graphs of the six trigonometric functions with the real
number domains introduced earlier. Also discussed are the domains, ranges, and periodic properties of these functions. The circular functions introduced in Section 5-2
will prove particularly useful in this regard.
It appears there is a lot to remember in this section. However, you only need to
be familiar with the graphs and properties of the sine, cosine, and tangent functions.
The reciprocal relationships discussed earlier will enable you to determine the graphs
and properties of the other three trigonometric functions from the graphs and properties of the sine, cosine, and tangent functions.
• Periodic Functions
Let’s return to circular points and wrapping functions discussed in Sections 5-1 and
5-2. Because the unit circle has a circumference of 2, we find that for a given value
of x (see Fig. 2 on the following page) we will return to the circular point
W(x) (a, b) if we add any integer multiple of 2 to x. Think of a point P moving
around the unit circle in either direction. Every time P covers a distance of 2, the
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circumference of the circle, it is back at the point where it started. Thus, for x any
real number,
sin (x 2k) sin x
k any integer
cos (x 2k) cos x
k any integer
b
FIGURE 2
a
b
P (cos x, sin x)
x units
arc length
r
1
(0, 1)
sin x
x rad
0
(1, 0)
(1, 0)
cos x
a
(0, 1)
Functions with this kind of repetitive behavior are called periodic functions. In
general, we have Definition 1.
DEFINITION 1
Periodic Functions
A function f is periodic if there exists a positive real number p such that
f (x p) f (x)
for all x in the domain of f. The smallest such positive p, if it exists, is called
the fundamental period of f (or often just the period of f ).
Both the sine and cosine functions are periodic with period 2.
• Graphs of
y sin x and
y cos x
We start by graphing
y sin x
x a real number
(1)
The graph of the sine function is the graph of the set of all ordered pairs of real numbers (x, y) that satisfy equation (1). Obtaining the graphs using point-by-point plotting is tedious and tends to obscure many important properties. We gain significantly
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Graphing Basic Trigonometric Functions
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more insight into the nature of these functions by observing how y sin x b varies
as the circular point P(a, b) moves around the unit circle.
We now know that the domain of the sine function is the set of all real numbers
R, the range is [1, 1], and the period is 2. Because the sine function has a period
of 2, we will concentrate on the graph over one period, from 0 to 2. Once we have
the graph for one period, we can complete as much of the rest of the graph as is
needed by repeating the graph to the left or to the right.
Figure 3 illustrates how y sin x b varies as x increases from 0 to 2 and
P(a, b) moves around the unit circle.
FIGURE 3 Variation in sin x.
/2
b
(0, 1)
a
b
P (cos x, sin x)
x
1
b
(1, 0)
a
0
2
(1, 0)
y sin x b
varies from:
As x
varies from:
a
0 to /2
/2 to 1 to
0
1 to
3/2 to 2
3/2
1
0 to 1
to 3/2
(0, 1)
0 to
0
To sketch a graph of y sin x over the interval [0, 2], we divide the interval
into four equal parts corresponding to the quadrants through which x varies and y
behaves uniformly. We choose as our basic unit on the x axis 2/4 /2. Of course,
all other real numbers are on the x axis, but for clarity we choose only to mark the
multiples of /2. To complete the sketch, we use the results in Figure 3 supplemented
where necessary by special real values (integer multiples of /6 or /4) or calculator values. The final graph is shown in Figure 4. The circle on the left—which is used
to define the sine function—is usually referred to mentally and is not part of the graph
of y sin x.
FIGURE 4 Graphing y sin x.
/2
b
(0, 1)
y
P(a, b)
1
x
1
b
b
(1, 0)
0
a
(0, 1)
a
(1, 0)
2
x
1
2
x
3
2
2
y sin x, 0 x 2
3/2
Figure 5 summarizes the above results and shows the sine graph over several
periods.
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5 Trigonometric Functions
FIGURE 5
Graph of y sin x
y
1
2
0
x
2
3
4
1
Period: 2
Domain: All real numbers
Symmetric with respect to the origin
Range: [1, 1]
If we proceed in the same way for the cosine function, we can obtain its graph.
Figure 6 shows how cos x a y varies as P(a, b) moves around the unit circle.
/2
b
FIGURE 6 Variation in cos x.
(0, 1)
a
b
P (cos x, sin x)
x
1
b
(1, 0)
0
a
As x
varies from:
a
(1, 0)
2
y cos x a
varies from:
0 to /2
1 to
/2 to to 3/2
(0, 1)
1 to
0
0 to
1
3/2 to 2
3/2
0
0 to 1
We can use the results in Figure 6, the fact that the cosine function is periodic
with period 2, and special or calculator values where necessary to obtain Figure 7.
FIGURE 7
Graph of y cos x
y
1
2
0
x
2
3
1
Period: 2
Domain: All real numbers
Symmetric with respect to the y axis
Range: [1, 1]
4
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The basic characteristics of the sine and cosine graphs should be learned so that
the curves can be sketched quickly. In particular, you should be able to answer the
following questions:
1. What is the period of each function (how often does the graph repeat)?
2. Where are the x intercepts?
3. Where are the y intercepts?
4. How far does each curve deviate from the x axis?
5. Where do the high and low points occur?
6. What are the symmetry properties?
EXPLORE-DISCUSS 1
• Graphs of y tan x and y cot x
/2
b
P(a, b)
(0, 1)
x
1
b
(1, 0)
0
a
a
(1, 0)
2
(A) Discuss how the graphs of the sine and cosine functions are related.
(B) How would you shift and/or reflect the sine graph to obtain the cosine graph?
(C) Is the graph of either y sin (x /2) or y sin (x /2) the same as
the graph of y cos x? Explain in terms of shifts and/or reflections.
We first discuss the graph of y tan x. Then from this graph, because cot x 1/(tan x), we will be able to get the graph of y cot x using reciprocals of ordinates.
Figure 8 shows that whenever the circular point P(a, b) is on the horizontal axis
(that is, whenever x k, k an integer), then (a, b) (1, 0), and tan x b/a 0/(1) 0. Whenever P(a, b) is on the vertical axis [that is, when x (/2) k,
k an integer], then (a, b) (0, 1), and tan x b/a (1)/0 is not defined (the
tangent function is discontinuous).
The values of x such that P(a, b) is on the horizontal axis in Figure 8 are the
zeros for tan x, or the x intercepts for the graph of y tan x. Thus, we can write
x intercepts: k
(0, 1)
3/2
tan x b
a
k an integer
As a first step in graphing y tan x, we locate the x intercepts on the x axis as illustrated in Figure 9.
FIGURE 8 The unit circle and tan x.
FIGURE 9 Intercepts and asymptotes for y tan x.
y
2
5
2
1
3
2
2
0
1
2
2
3
2
5
2
x
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5 Trigonometric Functions
The values of x such that P(a, b) is on a vertical axis in Figure 8 are points of
discontinuity. As a second step in graphing y tan x, we draw dashed vertical lines
through these points of discontinuity as illustrated in Figure 9—the graph cannot cross
these lines. These dashed vertical lines, called asymptotes, are convenient guidelines
for sketching the graph of y tan x. The line x a is a vertical asymptote for the
graph of y f(x) if f(x) either increases or decreases without bound as x approaches
a from the left or from the right. Thus, we write
Vertical asymptotes: x k
2
k an integer
We next investigate the behavior of the graph of y tan x in more detail between
the two asymptotes nearest the origin, that is, over the interval (/2, /2). Since
tan (x) tan x (Section 5-2), we only need to develop the graph for the interval
[0, /2), then we can reflect this graph through the origin to obtain the graph for the
entire interval (/2, /2).
Two points on the graph for the interval [0, /2) are easy to compute: tan 0 0 and tan (/4) 1. What happens to tan x as x approaches /2 from the left? If x
approaches /2 from the left, the circular point P(a, b) in Figure 9 stays in the first
quadrant, and a approaches 0 through positive values and b approaches 1. What happens to y tan x in the process? The calculator experiment in Example 1 will help
determine an answer.
EXAMPLE 1
Calculator Experiment
Form a table of values for y tan x approaching /2 1.570 796 through values
less than /2, starting at 0. Conclusion?
Solution
A table is created as follows:
x
0
0.5
1
1.57
1.5707
1.570 796
tan x
0
0.5
1.6
1,256
10,381
3,060,022
As x approaches /2 from the left, tan x appears to increase without bound.
Matched Problem 1
Form a table of values for y tan x with x approaching /2 1.570 796 through
values greater than /2, starting at 0. That is, use the negative of the x values used
in Example 1. Conclusion?
Figure 10(a) shows the graph resulting from the Example 1 analysis. The graph
can be completed for the interval (/2, /2) by reflecting the graph in Figure 10(a)
through the origin. Figure 10(b) shows the result.
5-6
FIGURE 10 Graph of y tan x,
/2 x /2.
Graphing Basic Trigonometric Functions
y
397
y
Vertical
asymptote
1
2
1
0
x
2
1
2
0
1
x
2
Vertical
asymptote
(a)
(b)
Proceeding in the same way for the other intervals between asymptotes, it appears
that the tangent function is periodic with period . To verify this, return again to Figure 8. If (a, b) are the coordinates of the circular point associated with x, then, using
symmetry of the unit circle and congruent reference triangles, (a, b) are the coordinates of the circular point associated with x . Hence,
tan (x ) b b
tan x
a a
and we conclude that the tangent function is periodic with period . In general,
tan (x k) tan x
k an integer
for all values of x for which both sides of the equation are defined.
To complete the graph of y tan x we need only to repeat the graph in Figure
10 to the left and right over intervals of to produce as much of the general graph
as we need (see Fig. 11). The main characteristics of the graph of the tangent function should be learned so that the graph can be sketched quickly.
Graph of y tan x
y
2
5
2
FIGURE 11
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
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We now turn to the cotangent function. Since cot x 1/tan x, we can graph y cot x by taking reciprocals of the y values in the graph of y tan x in Figure 11.
Note that the x intercepts and the vertical asymptotes are interchanged. The graph of
y cot x is shown in Figure 12. As with the tangent function, its main characteristics should be learned so that its graph can be sketched quickly.
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 12
EXPLORE-DISCUSS 2
• Graphs of
y csc x and
y sec x
(A) Discuss how the graphs of the tangent and cotangent functions are related.
(B) How would you shift and/or reflect the tangent graph to obtain the cotangent
graph?
(C) Is the graph of either y tan (x /2) or y tan (x /2) the same as
the graph of y cot x? Explain in terms of shifts and/or reflections.
Just as we obtained the graph of y cot x by taking reciprocals of the y values in
the graph of y tan x, since
csc x 1
sin x
and
sec x 1
cos x
we can obtain the graphs of y csc x and y sec x by taking reciprocals of the y
values in the graphs of y sin x and y cos x, respectively. Vertical asymptotes
occur at the x intercepts of either sin x or cos x.
The graphs of y csc x and y sec x are shown in Figures 13 and 14, respectively. As a graphing aid, we sketch in broken lines of y sin x and y cos x first
and then draw vertical asymptotes through the x intercepts. Check a few points on
the graphs with a calculator.
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Graphing Basic Trigonometric Functions
399
Graph of y csc x
y
y csc x 1
sin x
y sin x
2
3
2
2
3
2
1
0
1
2
x
2
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
FIGURE 13
Graph of y sec x
y
y sec x y cos x
1
2
3
2
2
0
1
2
3
2
x
2
1
cos 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
FIGURE 14
We have completed the discussion of the graphs of the six basic trigonometric
functions and their fundamental properties. In all cases we proceeded from basic definitions and properties of these functions. You should be able to sketch any of these
graphs and describe their fundamental attributes. Keeping the unit circle in mind
should prove helpful.
• Graphs on a
Graphing Utility
Now that you know how the graphs of the six trigonometric functions are generated
from basic definitions and properties, we turn to their graphs on a graphing utility,
which can produce these graphs almost instantaneously.
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5 Trigonometric Functions
EXAMPLE 2
Trigonometric Graphs on a Graphing Utility
Use a graphing utility to graph the functions
y sin x
y tan x
y sec x
for 2 x 2, 5 y 5. Display each graph in a separate viewing window
using a “connected” mode.
Solution
First set the graphing utility in radian and connected modes. Next enter the following window parameters, using 6.3 as an approximation for 2:
Xmin 6.3
Xmax 6.3
Xscl 1
Ymin 5
Ymax 5
Yscl 1
Now enter each function and produce its graph as indicated in Figure 15.
FIGURE 15 Graphing utility graphs
in “connected” mode.
5
6.3
5
6.3
6.3
5
6.3
6.3
6.3
5
5
5
(a) y sin x
(b) y tan x
(c) y sec x
In Figure 15(b) and (c), it appears that the graphing utility has also drawn the
vertical asymptotes for these functions. This is not the case. Most graphing utilities
calculate points on a graph and connect these points with line segments. The last point
plotted to the left of an asymptote and the first point plotted to the right of the asymptote will usually have very large y coordinates. If these y coordinates have opposite
sign, then the utility will connect the two points with a line that is nearly vertical,
and the line has the appearance of an asymptote. The utility is not performing any
asymptote analysis, it is simply connecting points with straight lines. No harm is done
as long as you recognize this, and the visual effect is close to that produced with the
asymptotes drawn in. You can set a utility to plot points without straight-line connections (“dot” mode), as shown in Figure 16. Unless stated to the contrary, we will
graph in the connected mode.
FIGURE 16 Graphing utility graphs
in “dot” mode.
5
6.3
Matched Problem 2
5
6.3
6.3
5
6.3
6.3
6.3
5
5
5
(a) y sin x
(b) y tan x
(c) y sec x
Repeat Example 2 for (A) y cos x, (B) y cot x, and (C) y csc x. (Use connected mode.)
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Graphing Basic Trigonometric Functions
401
Answers to Matched Problems
1.
2.
x
0
0.5
1
1.57
1.5707
1.570 796
tan x
0
0.5
1.6
1,256
10,381
3,060,022
As x approaches /2 from the right, tan x appears to decrease without bound.
(A) y cos x
(B) y cot x
(C) y csc x
5
6.3
5
6.3
6.3
5
EXERCISE
5
6.3
5
6.3
6.3
5
5-6
The figure below will be useful in many of the problems in
this exercise.
/2
b
(0, 1)
0
a
8. Which of the six basic trigonometric functions are undefined at x /2?
x
b
(1, 0)
7. Which of the six basic trigonometric functions are undefined at x 0?
a
b
P (cos x, sin x)
1
a
(1, 0)
6. What are the x intercepts for the graph of each function
over the interval 2 x 2?
(A) y cos x
(B) y tan x
(C) y sec x
2
(0, 1)
3/2
A
Answer Problems 1–12 without looking back in the text or
using a calculator. You can refer to the above figure.
1. What are the periods of the sine, cotangent, and cosecant
functions?
2. What are the periods of the cosine, tangent, and secant
functions?
3. How far does the graph of each function deviate from the
x axis?
(A) y cos x
(B) y tan x
(C) y csc x
4. How far does the graph of each function deviate from the
x axis?
(A) y sin x
(B) y cot x
(C) y sec x
5. What are the x intercepts for the graph of each function over
the interval 2 x 2?
(A) y sin x
(B) y cot x
(C) y csc x
B
9. At what values of x, 2 x 2, do the vertical asymptotes for the following functions cross the x axis?
(A) y cos x
(B) y tan x
(C) y csc x
10. At what values of x, 2 x 2, do the vertical asymptotes for the following functions cross the x axis?
(A) y sin x
(B) y cot x
(C) y sec x
11. Sketch graphs of each of the following functions over the
interval [2, 2]. Indicate the scale on the x axis in terms
of , and draw in vertical asymptotes using dashed lines
where appropriate.
(A) y cos x
(B) y tan x
(C) y csc x
12. Sketch graphs of each of the following functions over the
interval [2, 2]. Indicate the scale on the x axis in terms
of , and draw in vertical asymptotes using dashed lines
where appropriate.
(A) y sin x
(B) y cot x
(C) y sec x
13. (A) Describe a shift and/or reflection that will transform
the graph of y csc x into the graph of y sec x.
(B) Is either the graph of y csc (x /2) or y csc (x /2) the same as the graph of y sec x?
Explain in terms of shifts and/or reflections.
14. (A) Describe a shift and/or reflection that will transform
the graph of y sec x into the graph of y csc x.
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5 Trigonometric Functions
(B) Is either the graph of y sec (x /2) or y sec (x /2) the same as the graph of y csc x?
Explain in terms of shifts and/or reflections.
Problems 15–20 require the use of a graphing utility. These
problems offer a preliminary investigation into the relationships of the graphs of y sin x and y cos x with the
graphs of y A sin x, y A cos x, y sin Bx, y cos Bx,
y sin(x C), and y cos (x C). This important topic
is discussed in detail in the next section.
15. (A) Graph y A cos x (2 x 2, 3 y 3) for
A 1, 2, and 3, all in the same viewing window.
(B) Do the x intercepts change? If so, where?
(C) How far does each graph deviate from the x axis? (Experiment with additional values of A.)
(D) Describe how the graph of y cos x is changed by
changing the values of A in y A cos x.
16. Repeat Problem 15, replacing each occurrence of cosine
(cos) by sine (sin).
17. (A) Graph y sin Bx ( x , 2 y 2) for
B 1, 2, and 3, all in the same viewing window.
(B) How many periods of each graph appear in this viewing window? (Experiment with additional positive integer values of B.)
(C) Based on the observations in part B, how many periods
of the graph of y sin nx, n a positive integer, would
appear in this viewing window?
18. Repeat Problem 17, replacing each occurrence of sine (sin)
by cosine (cos).
19. (A) Graph y cos (x C) (2 x 2, 1.5 y 1.5) for C 0, /2, and /2, all in the same viewing
window. (Experiment with additional values of C.)
(B) Describe how the graph of y cos x is changed by
changing the values of C in y cos (x C).
20. Repeat Problem 19, replacing each occurrence of cosine
(cos) by sine (sin).
C
21. Try to calculate each of the following on your calculator.
Explain the results.
(A) sec (/2)
(B) tan (/2)
(C) cot ()
Problems 25 and 26 require the use of a graphing utility.
25. Graph f(x) sin x and g(x) x in the same viewing window (1 x 1, 1 y 1).
(A) What do you observe about the two graphs when x is
close to 0, say 0.5 x 0.5?
(B) Complete the table to three decimal places (use the
table feature on your graphing utility if it has one):
0.3
x
0.2
0.1
0.0
0.1
0.2
0.3
sin x
(In applied mathematics certain derivations, formulas and
calculations are simplified by replacing sin x with x for
small x.)
26. Graph h(x) tan x and g(x) x in the same viewing window (1 x 1, 1 y 1).
(A) What do you observe about the two graphs when x is
close to 0, say 0.5 x 0.5?
(B) Complete the table to three decimal places (use the
table feature on your graphing utility if it has one):
0.3
x
0.2
0.1
0.0
0.1
0.2
0.3
tan x
(In applied mathematics certain derivations, formulas and
calculations are simplified by replacing tan x with x for
small x.)
In Problems 27–34, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
27. If functions f and g are periodic with the same period, then
the sum f g and difference f g are also periodic.
28. If functions f and g are periodic with the same period, then
the product fg and quotient f/g are also periodic.
29. If f is any function and g is periodic, then the composite f°g
is periodic.
30. If f is periodic and g is any function, then the composite f°g
is periodic.
31. If function f is periodic, then the function g defined by g(x)
5f (x) is periodic with the same period as f.
22. Try to calculate each of the following on your calculator.
Explain the results.
(A) csc (B) tan (/2)
(C) cot 0
32. If function f is periodic, then the function h defined by h(x)
f (5 x) is periodic with the same period as f.
23. Explain why a function that is increasing throughout its domain cannot be periodic.
33. If function f is periodic, then the function j defined by j(x)
f (5x) is periodic with the same period as f.
24. Explain why a nonconstant polynomial function cannot be
periodic.
34. If function f is periodic, then the function k defined by k(x)
5 f(x) is periodic with the same period as f.