387
5-6 Graphing Basic Trigonometric Functions
51. Geometry. Find r in the accompanying figure (to two significant digits) so that the circle is tangent to all three sides
of the isosceles triangle. [Hint: The radius of a circle is
perpendicular to a tangent line at the point of tangency.]
★★
52. Geometry. Find r in the accompanying figure (to two significant digits) so that the smaller circle is tangent to the
larger circle and the two sides of the angle. [See the hint in
Problem 51.]
r
30⬚
2.0 meters
r
2.0 in.
30⬚
Section 5-6 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
8
7
6
5
F
A
J
A
O
D
F
A
J
A
Month
Sunrise times and time of year
(a)
O
D
Pressure at eardrum
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.
Sunrise time
★★
1
600
1
300
Time (seconds)
Soundwave arriving at eardrum
(b)
FIGURE 1
Periodic phenomena.
In this section we discuss the graphs of the six trigonometric functions with
real number domains introduced earlier. We also discuss 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.
5 TRIGONOMETRIC 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) 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 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
Functions with this kind of repetitive behavior are called periodic functions.
In general, we have Definition 1.
FIGURE 2
b
a
b
P (cos x, sin x)
x units
(arc length)
1
(0, 1)
r⫽
388
x rad
0
(⫺1, 0)
cos x
sin x
(1, 0)
a
(0, ⫺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 ⴝ sin x and y ⴝ cos x
We start by graphing
y ⫽ sin x
x a real number
(1)
5-6 Graphing Basic Trigonometric Functions
389
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 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.
␲/2
b
FIGURE 3
Variation in sin x.
(0, 1)
a
b
P (cos x, sin x)
x
1
b
␲
(⫺1, 0)
a
0
As x
varies from
y ⴝ sin x ⴝ b
varies from
0 to ␲/2
␲/2 to ␲
␲ to 3␲/2
3␲/2 to 2␲
0 to ⫺1
1 to ⫺0
0 to ⫺1
⫺1 to ⫺0
a
2␲
(1, 0)
(0, ⫺1)
3␲/2
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.
␲/2
b
FIGURE 4
Graphing y ⫽ sin x.
(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 ⱕ x ⱕ 2␲
3␲/2
Figure 5 summarizes the above results and shows the sine graph over several
periods.
390
5 TRIGONOMETRIC FUNCTIONS
FIGURE 5
GRAPH OF y ⴝ 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]
A graphing utility can be used to interactively explore the relationship between
the unit circle definition of the sine function and the graph of the sine function
as shown in Figure 4. Explore/Discuss 1 provides the details.
Explore/Discuss
1
Set your graphing utility in radian and parametric modes. (Parametric
equations will be covered in detail in Sec. 10-5.) Make the entries as
indicated in Figure 6 to obtain the indicated graph (2␲ is entered as
Tmax and Xmax, ␲/2 is entered as Xscl).
FIGURE 6
Use TRACE and move back and forth between the unit circle and the
graph of the sine function for various values of T as T increases from 0
to 2␲. Discuss what happens in each case. Figure 7 illustrates the case
for T ⫽ 0.
FIGURE 7
Repeat the exploration with Y2T ⫽ cos (T)
391
5-6 Graphing Basic Trigonometric Functions
If we proceed in the same way for the cosine function as we did for the sine
function, we can obtain its graph. Figure 8 shows how cos x ⫽ a ⫽ y varies as
P(a, b) moves around the unit circle.
␲/2
b
FIGURE 8
Variation in cos x.
(0, 1)
a
b
P (cos x, sin x)
b
␲
(⫺1, 0)
0
a
y ⴝ cos x ⴝ a
varies from
As x
varies from
x
1
a
(1, 0)
2␲
0 to ␲/2
␲/2 to ␲
␲ to 3␲/2
3/␲ to 2␲
(0, ⫺1)
1 to ⫺0
0 to ⫺1
⫺1 to ⫺0
0 to ⫺1
3␲/2
We can use the results in Figure 8, the fact that the cosine function is periodic with period 2␲, and special or calculator values where necessary to obtain
Figure 9.
FIGURE 9
GRAPH OF y ⴝ cos x
y
1
⫺2␲
⫺␲
0
␲
x
2␲
3␲
4␲
⫺1
Period: 2␲ Domain: All real numbers
Symmetric with respect to the y axis
Range: [⫺1, 1]
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?
392
5 TRIGONOMETRIC FUNCTIONS
Explore/Discuss
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 either the graph of y ⫽ sin(x ⫺ ␲/2) or that of y ⫽ sin(x ⫹ ␲/2)
the same as the graph of y ⫽ cos x? Explain in terms of shifts
and/or reflections.
Graphs of y ⴝ tan x and y ⴝ cot x
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 10 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).
␲/2
b
FIGURE 10
The unit circle and tan x.
P(a, b)
(0, 1)
x
1
b
␲
(⫺1, 0)
0
a
a
(1, 0)
2␲
(0, ⫺1)
3␲/2
tan x ⫽
b
a
The values of x such that P(a, b) is on the horizontal axis in Figure 10 are
the zeros for tan x, or the x intercepts for the graph of y ⫽ tan x. Thus, we can
write
x intercepts: k␲
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 11 at the top of the next page.
The values of x such that P(a, b) is on a vertical axis in Figure 10 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 11—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
5-6 Graphing Basic Trigonometric Functions
393
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
x⫽
Vertical asymptotes:
FIGURE 11
␲
⫹ k␲
2
k an integer
y
Intercepts and asymptotes for
y ⫽ tan x.
⫺2␲
1
⫺␲
␲
⫺
2
3␲
⫺
2
5␲
⫺
2
␲
0
⫺1
␲
2
2␲
3␲
2
5␲
2
x
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 10 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
Calculator Experiment
1
Solution
Form a table of values for y ⫽ tan x with x approaching ␲/2 ⬇ 1.570 796
through values less than ␲/2, starting at 0. Conclusion?
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 approaching ⫺␲/2 ⬇ ⫺1.570 796 through
values greater than ⫺␲/2, starting at 0. That is, use the negatives of the x values
used in Example 1. Conclusion?
394
5 TRIGONOMETRIC FUNCTIONS
Figure 12(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 12(a) through the origin. Figure 12(b) shows the result.
FIGURE 12
y
Graph of y ⫽ tan x, ⫺␲/2 ⬍ x ⬍
␲/2.
y
Vertical
asymptote
1
⫺
␲
2
1
0
⫺1
x
␲
2
⫺
0
␲
2
x
␲
2
⫺1
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 10. 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 ⴙ k␲) ⴝ 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 12 to the left and right over intervals of ␲ to produce as much of the general
graph as we need (see Fig. 13). The main characteristics of the graph of the tangent function should be learned so that the graph can be sketched quickly.
FIGURE 13
GRAPH OF y ⴝ tan x
y
⫺2␲
5␲
⫺
2
1
⫺␲
3␲
⫺
2
␲
⫺
2
␲
0
⫺1
␲
2
2␲
3␲
2
5␲
2
x
395
5-6 Graphing Basic Trigonometric Functions
FIGURE 13
(continued)
Period:
␲
Domain:
All real numbers except ␲/2 ⫹ k␲, k an integer
Range:
All real numbers
Symmetric with respect to the origin
Increasing function between asymptotes
Discontinuous at x ⫽ ␲/2 ⫹ k␲, k an integer
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 13. Note that the x intercepts and the vertical asymptotes are interchanged.
The graph of y ⫽ cot x is shown in Figure 14. As with the tangent function, its
main characteristics should be learned so that its graph can be sketched quickly.
FIGURE 14
GRAPH OF y ⴝ 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 asymptotes
Discontinuous at x ⫽ k␲, k an integer
Explore/Discuss
3
(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 either the graph of 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.
396
5 TRIGONOMETRIC FUNCTIONS
Graphs of y ⴝ csc x and y ⴝ sec x
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
sec x ⫽
and
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 15 and 16, 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.
FIGURE 15
GRAPH OF y ⴝ 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 ⱕ ⫺1 or y ⱖ 1
Symmetric with respect to the origin
Discontinuous at x ⫽ k␲, k an integer
5-6 Graphing Basic Trigonometric Functions
FIGURE 16
397
GRAPH OF y ⴝ sec x
y
y ⫽ sec x ⫽
1
cos x
y ⫽ cos x
1
⫺2␲
⫺
3␲
2
⫺␲
⫺
␲
2
0
⫺1
␲
2
␲
3␲
2
x
2␲
Period:
2␲
Domain:
All real numbers except ␲/2 ⫹ k␲, k an integer
Range:
All real numbers y such that y ⱕ ⫺1 or y ⱖ 1
Symmetric with respect to the y axis
Discontinuous at x ⫽ ␲/2 ⫹ k␲, k an integer
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.
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␲ ⱕ x ⱕ 2␲, ⫺5 ⱕ y ⱕ 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
398
5 TRIGONOMETRIC FUNCTIONS
Now enter each function and produce its graph as indicated in Figure 17.
5
5
⫺6.3
6.3
⫺6.3
⫺5
(a) y ⫽ sin x
5
6.3
⫺6.3
6.3
⫺5
⫺5
(b) y ⫽ tan x
(c) y ⫽ sec x
FIGURE 17
Graphing utility graphs in “connected” mode.
In Figures 17(b) and 17(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 18.
Unless stated to the contrary, we will graph in the connected mode.
5
5
⫺6.3
6.3
⫺6.3
⫺5
(a) y ⫽ sin x
5
6.3
⫺6.3
6.3
⫺5
⫺5
(b) y ⫽ tan x
(c) y ⫽ sec x
FIGURE 18
Graphing utility graphs in “dot” mode.
MATCHED PROBLEM
2
Repeat Example 2 for (A) y ⫽ cos x, (B) y ⫽ cot x, and (C) y ⫽ csc x. (Use
connected mode.)
Answers to Matched Problems
1.
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.
5-6 Graphing Basic Trigonometric Functions
2. (A) y ⫽ cos x
(B) y ⫽ cot x
5
(C) y ⫽ csc x
5
⫺6.3
5
⫺6.3
6.3
⫺5
399
6.3
⫺6.3
⫺5
6.3
⫺5
6. What are the x intercepts for the graph of each function
over the interval ⫺2␲ ⱕ x ⱕ 2␲?
EXERCISE 5-6
The figure below will be useful in many of the problems in this
exercise.
␲/2
b
(0, 1)
0
a
(A) y ⫽ sin x
a
(1, 0)
(C) y ⫽ sec x
(B) y ⫽ tan x
(C) y ⫽ csc x
8. For what values of x, ⫺2␲ ⱕ x ⱕ 2␲, are the following
functions not defined?
x
b
(B) y ⫽ tan x
7. For what values of x, ⫺2␲ ⱕ x ⱕ 2␲, are the following
functions not defined?
(A) y ⫽ cos x
a
b
P (cos x, sin x)
1
␲
(⫺1, 0)
(A) y ⫽ cos x
2␲
(C) y ⫽ sec x
B
9. At what points, ⫺2␲ ⱕ x ⱕ 2␲, do the vertical asymptotes for the following functions cross the x axis?
(A) y ⫽ cos x
(0, ⫺1)
(B) y ⫽ cot x
(B) y ⫽ tan x
(C) y ⫽ csc x
10. At what points, ⫺2␲ ⱕ x ⱕ 2␲, do the vertical asymptotes for the following functions cross the x axis?
3␲/2
Figure for Problems 1–12.
(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
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␲ ⱕ x ⱕ 2␲?
(A) y ⫽ sin x
(B) y ⫽ cot x
(C) y ⫽ csc x
(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.
(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.
400
5 TRIGONOMETRIC FUNCTIONS
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.
20. (A) Graph y ⫽ sin (x ⫹ C), (⫺2␲ ⱕ x ⱕ 2␲, ⫺1.5 ⱕ y ⱕ
1.5), for C ⫽ 0, ⫺␲/2, and ␲/2, all in the same viewing window. (Experiment with additional values of C.)
15. (A) Graph y ⫽ A cos x, (⫺2␲ ⱕ x ⱕ 2␲, ⫺3 ⱕ y ⱕ 3),
for A ⫽ 1, 2, and ⫺3, all in the same viewing
window.
C
(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. (A) Graph y ⫽ A sin x, (⫺2␲ ⱕ x ⱕ 2␲, ⫺3 ⱕ y ⱕ 3),
for A ⫽ 1, 3, and ⫺2, 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 ⫽ sin x is changed by
changing the values of A in y ⫽ A sin x?
17. (A) Graph y ⫽ sin Bx (⫺␲ ⱕ x ⱕ ␲, ⫺2 ⱕ y ⱕ 2), for
B ⫽ 1, 2, and 3, all in the same viewing window.
(B) How many periods of each graph appear in this
viewing rectangle? (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. (A) Graph y ⫽ cos Bx (⫺␲ ⱕ x ⱕ ␲, ⫺2 ⱕ y ⱕ 2), for
B ⫽ 1, 2, and 3, all in the same viewing window.
(B) How many periods of each graph appear in this
viewing rectangle? (Experiment with additional
positive integer values of B.)
(C) Based on the observations in part B, how many
periods of the graph of y ⫽ cos nx, n a positive
integer, would appear in this viewing window?
19. (A) Graph y ⫽ cos (x ⫹ C), (⫺2␲ ⱕ x ⱕ 2␲, ⫺1.5 ⱕ y ⱕ
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)?
(B) Describe how the graph of y ⫽ sin x is changed by
changing the values of C in y ⫽ sin (x ⫹ C)?
21. Try to calculate each of the following on your calculator.
Explain the results.
(A) sec (␲/2)
(B) tan (⫺␲/2)
(C) cot (⫺␲)
22. Try to calculate each of the following on your calculator.
Explain the results.
(A) csc ␲
(B) tan (␲/2)
(C) cot 0
Problems 23 and 24 require the use of a graphing utility.
23. Graph f(x) ⫽ sin x and g(x) ⫽ x in the same viewing window (⫺1 ⱕ x ⱕ 1, ⫺1 ⱕ y ⱕ 1).
(A) What do you observe about the two graphs when x is
close to 0, say ⫺0.5 ⱕ x ⱕ 0.5?
(B) Complete the table to three decimal places (use the
table feature on your graphing utility if it has one):
x
⫺0.3
⫺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 values of x .)
ⱍⱍ
24. Graph h(x) ⫽ tan x and g(x) ⫽ x in the same viewing window (⫺1 ⱕ x ⱕ 1, ⫺1 ⱕ y ⱕ 1).
(A) What do you observe about the two graphs when x is
close to 0, say ⫺0.5 ⱕ x ⱕ 0.5?
(B) Complete the table to three decimal places (use the
table feature on your graphing utility if it has one):
x
⫺0.3
⫺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 values of x .)
ⱍⱍ