1 Graphs of y = a sin x and y = a cos x
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298
CHAPTER 10 Graphs of the Trigonometric Functions
Table 10.1
Key
values
y = sin x
y = cos x
0
0
1
p
2
1
0
p
0
3p
2
-1
-1
0
2p
0
1
Basic Features of the Graphs of y = sin x and y = cos x
1. The domain is all values of x.
2. The range is -1 … y … 1, or 3 -1, 14 .
3. The amplitude (half the distance between the maximum value and the minimum value) is 1.
4. Both graphs are exactly the same shape (called sinusoidal).
5. The graph of the cosine curve is shifted p>2 units to the left of the sine curve.
6. For both graphs, the values of y repeat every 2p units of x. We therefore say
that the functions are periodic with period 2p.
7. The functions have zeros, maximum values, and minimum values when x is a
multiple of p>2. The behaviour from 0 to 2p at these key values is summarized in Table 10.1.
y
Period = 2p
1
Amplitude
0
y = sin x
p
p
2
y = cos x
2p
3p
2
To obtain the graph of y = a sin x, note that all the y-values obtained for the graph of
y = sin x are to be multiplied by the number a. In this case, the greatest value of the sine
function is 0 a 0 , and the curve will have no value less than - 0 a 0 . Therefore, the range is
3 - 0 a 0 , 0 a 0 4 , and the amplitude of the curve is 0 a 0 . This is also true for y = a cos x.
x
E X A M P L E 1 Plotting the graph of y = a sin x
−1
p
2
shift
Plot the graph of y = 2 sin x.
Since a = 2, the amplitude of this curve is 0 2 0 = 2. This means that the maximum
value of y is 2 and the minimum value is y = -2. The table of values follows, and the
curve is shown in Fig. 10.6.
Fig. 10.5
y
2
Amplitude
0
p
2
2p
3p
2
p
−2
0
p
6
p
3
p
2
2p
3
5p
6
p
y
0
1
1.73
2
1.73
1
0
x
x
7p
6
4p
3
3p
2
5p
3
11p
6
2p
y
-1
-1.73
-2
-1.73
-1
0
■
E X A M P L E 2 Plotting the graph of y = a cos x
Plot the graph of y = -3 cos x.
In this case, a = -3, and this means that the amplitude is 0 -3 0 = 3. Therefore, the
maximum value of y is 3, and the minimum value of y is -3. The table of values follows, and the curve is shown in Fig. 10.7.
Fig. 10.6
y
3
x
0
p
6
p
3
p
2
2p
3
5p
6
p
y
-3
-2.6
-1.5
0
1.5
2.6
3
x
7p
6
4p
3
3p
2
5p
3
11p
6
2p
y
2.6
1.5
0
-1.5
-2.6
-3
2
Amplitude
1
0
−1
x
p
2
p
−2
−3
Fig. 10.7
3p
2
2p
x
Note that the effect of the negative sign with the number a is to invert the curve about
the x-axis.
■
Apart from the fact that the range of the functions y = a sin x and y = a cos x is
3 - 0 a 0 , 0 a 0 4 , we can see from the previous examples that the number a has no other
effect on the basic features of these functions as compared to those of the functions
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299
10.1 Graphs of y = a sin x and y = a cos x
Table 10.2
Key
values
0
y = a sin x y = a cos x
0
a
p
2
a
0
p
0
-a
3p
2
-a
0
2p
0
a
y = sin x and y = cos x. In particular, they have the same sinusoidal shape and the
same period, and their zeros, maximum points, and minimum points are located at the
same key values as before (see Table 10.2). It follows that by knowing the basic features of the sine and cosine functions, we can sketch the graphs of functions of the form
y = a sin x and y = a cos x quickly by simply using the appropriate amplitude and
inverting the curve when necessary.
E X A M P L E 3 Using key values to sketch a graph
Sketch the graph of y = 40 cos x.
First, we set up a table of values for the points where the curve has its zeros, maximum points, and minimum points:
y
Max.
40 Max.
20
0
−20
0
p
2
p
y
40
max.
0
-4 0
min.
Zeros
p
2
−40
2p
3p
2
p
3p
2
x
2p
0
40
max.
x
Now, we plot these points and join them, knowing the basic sinusoidal shape of the
curve. See Fig. 10.8.
■
Min.
E X A M P L E 4 Using key values to sketch a graph
Sketch the graph of y = -2 sin x.
The key values between 0 and 2p are the following:
Fig. 10.8
Practice Exercise
1. For the graph of y = -6 sin x, set up a
table of key values for 0 … x … 2p.
y
0
0
p
2
p
y
0
-2
min.
0
3p
2
2p
2
max.
0
y = −2 sin x
2
− 5p
2
x
p
2
3p
2
5p
2
x
5p
The graph from x = - 5p
2 to x = 2 is shown in Fig. 10.9, plotted in the
same set of axes as the function y = sin x. The effect of the constant
a = -2 in terms of the change in amplitude and the inversion of the curve
can be readily seen.
■
y = sin x
−2
Fig. 10.9
E XE RC IS ES 1 0 .1
In Exercises 1 and 2, graph the function if the given changes are made
in the indicated examples of this section.
1. In Example 2, if the sign of the coefficient of cos x is changed,
plot the graph of the resulting function.
2. In Example 4, if the sign of the coefficient of sin x is changed,
display the graph of the resulting function.
In Exercises 3–6, complete the following table for the given functions
and then plot the resulting graphs.
x
-p
- 3p
4
- p2
- p4
0
p
4
p
2
3p
4
p
y
x
5p
4
3p
2
7p
4
2p
9p
4
5p
2
11p
4
3p
y
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300
CHAPTER 10 Graphs of the Trigonometric Functions
3. y = sin x
4. y = cos x
5. y = 3 cos x
6. y = -4 sin x
31. The graph displayed on an oscilloscope can be represented by
y = -0.05 sin x. Display this curve on a graphing calculator.
32. The displacement y (in cm) of the end of a robot arm for welding
is y = 4.75 cos t, where t is the time (in s). Display this curve on
a graphing calculator.
In Exercises 7–22, sketch the graphs of the given functions.
8. y = 5 sin x
7. y = 3 sin x
9. y =
5
2
In Exercises 33–36, the graph of a function of the form y = a sin x or
y = a cos x is shown. Determine the specific function of each.
10. y = 35 sin x
sin x
11. y = 200 cos x
12. y = 0.25 cos x
13. y = 0.8 cos x
14. y =
15. y = -sin x
16. y = - 300 sin x
3
2
33.
cos x
17. y = -1500 sin x
18. y = - 0.2 sin x
19. y = -cos x
20. y = - 8 cos x
21. y = -50 cos x
22. y = - 0.4 cos x
24. y = - 30 sin x
25. y = 12 cos x
26. y = 2 cos x
28. Find the function and graph it for a function of the form y = a sin x
that passes through 13p>2, - 22.
29. Find the function and graph it for a function of the form y = a cos x
that passes through 1p, 22.
30. Find the function and graph it for a function of the form y = a cos x
that passes through 12p, - 22.
x
2p
0
−0.2
35.
36.
y
y
1.5
6
2p
0
x
x
2p
0
−6
−1.5
In Exercises 37–40, find the value of a for either y = a sin x or
y = a cos x, whichever is correct, such that the given point is on the
graph. The amplitude of each function is 2.50. Thereby determine the
function. (All points are located such that the x value is between - p
and p.)
37. 10.67, -1.552
38. 1 - 1.20, 0.902
40. 1 - 2.47, - 1.552
39. (2.07, 1.20)
Answers to Practice Exercise
1.
x
0
p
2
p
3p
2
2p
y
0
-6
min.
0
6
max.
0
Graphs of y = a sin bx and y = a cos bx
10.2
We learned in Section 10.1 that the values of the functions y = a sin x and y = a cos x
repeat every 2p units of x, making them periodic with period 2p. More generally, we
say that a function F has period P if F1x2 = F1x + P2 for all x in the domain of F,
and P is the smallest such number. In other words, the period is the x-distance between
a point and the next corresponding point after which the values of y repeat.
Let us now plot the curve y = sin 2x. This means that we choose a value of x, multiply this value by 2, and find the sine of the result. This leads to the following table of
values for this function:
1FSJPEPGB'VODUJPO t (SBQITPG
y = a sin bx and y = a cos bx t *NQPSUBOU
7BMVFTGPS4LFUDIJOH t $ZDMF
y
1
−1
x
−4
27. Find the function and graph it for a function of the form y = a sin x
that passes through 1p>2, - 22.
0
2p
0
In Exercises 27–32, solve the given problems.
y
0.2
4
Although units of p are convenient, we must remember that p is only
a number. Numbers that are not multiples of p may be used. In
Exercises 23–26, plot the indicated graphs by finding the values of y
that correspond to values of x of 0, 1, 2, 3, 4, 5, 6, and 7 on a
calculator. (Remember, the numbers 0, 1, 2, and so on represent
radian measure.)
23. y = sin x
34.
y
p
4
3p
4
p
2
Period = p
Fig. 10.10
p
5p
4
x
p
4
p
2
3p
8
p
2
5p
8
3p
4
7p
8
p
9p
8
5p
4
0
p
8
p
4
3p
4
p
5p
4
3p
2
7p
4
2p
9p
4
5p
2
0
0.7
1
0.7
0
-0.7
-1
-0.7
0
0.7
1
x
0
2x
y
Plotting these points, we have the curve shown in Fig. 10.10.
From the table and Fig. 10.10, note that y = sin 2x repeats after p units of x. The
effect of the 2 is that the period of y = sin 2x is half the period of the curve of y = sin x.
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10.2 Graphs of y = a sin bx and y = a cos bx
LEARNING TIP
E X A M P L E 1 Finding the period of a function
If the period of a function F 1x 2 is P,
then the period of F 1bx 2 is P>b. Since
each of the functions sin x and cos x
has a period of 2p, each of the functions sin bx and cos bx has a period of
2p>b.
Practice Exercise
p
(a) The period of cos 4x is 2p
4 = 2.
2p
(c) The period of sin 12x is 1 = 4p.
2
(b) The period of sin 3px is 2p
3p = 3 .
2p
(d) The period of cos p4 x is p = 8.
4
2
In (a), the period tells us that the curve of y = cos 4x will repeat every p>2 (approximately
1.57) units of x. In (b), we see that the curve of y = sin 3px will repeat every 2>3 of a unit.
In (c) and (d), the periods are longer than those of y = sin x and y = cos x.
■
Combining the value of the period with the value of the amplitude from Section
10.1, we conclude that the functions y = a sin bx and y = a cos bx have an amplitude
of 0 a 0 and a period of 2p>b. These properties are very useful in sketching these
functions.
Find the period of each function.
1. y = sin px
301
2. y = cos 13x
E X A M P L E 2 Sketching the graph of y = a sin bx
Sketch the graph of y = 3 sin 4x for 0 … x … p.
Since a = 3, the amplitude is 3. The 4x tells us that the period is 2p>4 = p>2.
This means that y = 0 for x = 0 and for y = p>2. Since this sine function is zero
halfway between x = 0 and x = p>2, we find that y = 0 for x = p>4. Also, the fact
that the graph of the sine function reaches its maximum and minimum values halfway
between zeros means that y = 3 for x = p>8, and y = -3 for x = 3p>8. Note that
the values of x in the following table are those for which 4x = 0, p>2, p, 3p>2, 2p,
and so on, which correspond to the key values listed in Tables 10.1 and 10.2.
y
3
Max.
0
−3
Max.
p
_
2
p
Min.
Min.
x
Period
Fig. 10.11
LEARNING TIP
By finding one-fourth of the period,
we can easily find the important values
for sketching the curve.
x
0
p
8
p
4
3p
8
p
2
5p
8
3p
4
7p
8
p
y
0
3
0
-3
0
3
0
-3
0
Using the values from the table and the fact that the curve is sinusoidal in form, we
sketch the graph of this function in Fig. 10.11. We see that the key values where the
function has zeros, maxima, and minima occur when x is a multiple of p>8, which is
exactly one-fourth of the period.
■
Note from Example 2 that an important distance in sketching a sine curve or a
cosine curve is one-fourth of the period. For y = a sin bx, it is one-fourth of the period
from the origin to the first value of x where y is at its maximum (or minimum) value.
Then we proceed another one-fourth period to a zero, another one-fourth period to the
next minimum (or maximum) value, another to the next zero (this is where the period is
completed), and so on.
Similarly, one-fourth of the period is useful in sketching the graph of y = cos bx.
For this function, the maximum (or minimum) value occurs for x = 0. At the following one-fourth-period values, there is a zero, a minimum (or maximum), a zero, and a
maximum (or minimum) at the start of the next period.
We now summarize the important values for sketching the graphs of y = a sin bx
and y = a cos bx.
Important Values for Sketching y = a sin bx and y = a cos bx
1. The amplitude: 0 a 0
2. The period: 2p>b
3. Values of the function for each one-fourth period