2 Graphs of y = a sin bx and y = a cos bx
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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
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302
CHAPTER 10 Graphs of the Trigonometric Functions
E X A M P L E 3 Using important values to sketch a graph
y
2
0
p
3
2p
3
p
4p
3
2p
5p
3
x
Sketch the graph of y = -2 cos 3x for 0 … x … 2p.
Note that the amplitude is 2 and the period is 2p
3 . This means that one-fourth of the
p
=
.
Since
the
cosine
curve
is
at
a maximum or minimum for x = 0,
period is 14 * 2p
3
6
we find that y = -2 for x = 0 (the negative value is due to the minus sign before the
function), which means it is a minimum point. The curve then has a zero at x = p6 , a
maximum value of 2 at x = 2 1 p6 2 = p3 , a zero at x = 3 1 p6 2 = p2 , and its next value of
- 2 at x = 4 1 p6 2 = 2p
3 , and so on. Therefore, we have the following table:
x
0
p
6
p
3
p
2
2p
3
5p
6
p
7p
6
4p
3
3p
2
5p
3
11p
6
2p
y
-2
0
2
0
-2
0
2
0
-2
0
2
0
-2
−2
Using this table and the sinusoidal shape of the cosine curve, we sketch the function in
Fig. 10.12.
■
Fig. 10.12
For a periodic function, a cycle is any section of the graph that includes exactly one
period. Fig. 10.11 shows two cycles, whereas Fig. 10.12 shows three.
E X A M P L E 4 Graph of y = a cos bx—application
A generator produces a voltage V = 200 cos 50pt, where t is the time in seconds (50p
is angular velocity, so it has units of rad>s; thus, 50pt is an angle in radians). Graph V
as a function of t for 0 … t … 0.06 s.
The amplitude is 200 V and the period is 2p> 150p2 = 0.04 s. Since one-fourth of
the period is 0.01 s, the function has zeros, maxima, and minima when x is a multiple of
0.01. Thus, we have the following table of key values:
V
200
0.06
t
t1s 2
−200
V1V 2
Fig. 10.13
0
0.01
0.02
0.03
0.04
0.05
0.06
200
0
-200
0
200
0
-200
The graph is shown in Fig. 10.13. Note that between 0 and 0.06 s, the function completes 1.5 cycles. We do not consider negative values of t, for they have no real meaning in this problem.
■
E XE R C I SES 1 0 .2
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 coefficient of x is changed from 4 to 6, sketch
the graph of the resulting function.
2. In Example 3, if the coefficient of x is changed from 3 to 4, sketch
the graph of the resulting function.
In Exercises 3–22, find the period of each function.
3. y = 2 sin 6x
4. y = 4 sin 2x
5. y = 3 cos 8x
6. y = 28 cos 10x
7. y = -2 sin 12x
8. y =
- 15
sin 5x
9. y = -cos 16x
10. y = -4 cos 2x
11. y = 520 sin 2px
12. y = 2 sin 3px
13. y = 3 cos 4px
14. y = 4 cos 10px
15. y = 15 sin
1
3x
16. y = - 25 sin
2
5x
17. y = - 12 cos 23x
19. y = 0.4 sin
2px
3
21. y = 3.3 cos p2x
18. y = 13 cos 14x
20. y = 1.5 cos
px
10
22. y = - 12.5 sin 2x
p
In Exercises 23–42, sketch the graphs of the given functions. (These
are the same functions as in Exercises 3–22.)
23. y = 2 sin 6x
24. y = 4 sin 2x
25. y = 3 cos 8x
26. y = 28 cos 10x
27. y = -2 sin 12x
28. y = - 15 sin 5x
29. y = -cos 16x
30. y = - 4 cos 2x
31. y = 520 sin 2px
32. y = 2 sin 3px
33. y = 3 cos 4px
34. y = 4 cos 10px
35. y = 15 sin 13x
36. y = - 25 sin 25x
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303
10.3 Graphs of y = a sin 1bx + c2 and y = a cos 1bx + c2
37. y = - 12 cos 23x
39. y = 0.4 sin
38. y =
2px
3
1
3
56. Find the function and graph it for a function of the form
y = -2 cos bx that passes through 1p>2, 22 and for which b has
the smallest possible positive value.
cos 14x
40. y = 1.5 cos
px
10
42. y = - 12.5 sin 2x
p
41. y = 3.3 cos p2x
In Exercises 43–46, the period is given for a function of the form
y = sin bx. Write the function corresponding to the given period.
p
2p
1
44.
45.
46. 6
3
5
3
In Exercises 47–50, graph the given functions. In Exercises 47 and 48,
first rewrite the function with a positive angle, and then graph the
resulting function. (Refer to Eq. 8.7 for trigonometric functions of
negative angles.)
43.
48. y = -5 cos 1 -4px2
47. y = 3 sin1 -2x2
49. y = 8 ͉ cos
1 p2 x 2 ͉
50. y = 0.4 ͉ sin 6x ͉
In Exercises 51–60, solve the given problems.
51. By noting the periods of sin 2x and sin 3x, find the minimum
period of the function y = sin 2x + sin 3x.
57. The standard electric voltage in a 60-Hz alternating-current circuit is given by V = 170 sin 120pt, where t is the time in seconds. Sketch the graph of V as a function of t for 0 … t … 0.05 s.
58. To tune the instruments of an orchestra before a concert, an A
note is struck on a piano. The piano wire vibrates with a displacement y (in mm) given by y = 3.2 cos 880pt, where t is in seconds. Sketch the graph of y vs. t for 0 … t … 0.01 s.
59. The velocity v (in cm>s) of a piston is v = 450 cos 3600t, where
t is in seconds. Sketch the graph of v vs. t for 0 … t … 0.006 s.
60. On a certain day in St. John, New Brunswick, the difference
between high tide and low tide was 6.4 m. The period was about
12.4 h. Find a cosine function that describes these tides if high
tide was at midnight.
In Exercises 61–64, the graph of a function of the form y = a sin bx
or y = a cos bx is shown. Determine the specific function of each.
61.
52. By noting the period of cos 12x and cos 13x, find the minimum
period of the function y = cos 12x + cos 13x.
53. Find the function and graph it for a function of the form
y = -2 sin bx that passes through 1p>4, -22 and for which b
has the smallest possible positive value.
0
54. Find the function and graph it for a function of the form
y = 2 sin bx that passes through 1p>6, 22 and for which b has
the smallest possible positive value.
62.
y
0.5
8
x
p
0
64.
y
y
4
55. Find the function and graph it for a function of the form
y = 2 cos bx that passes through 1p, 02 and for which b has the
smallest possible positive value.
x
p
4
−8
−0.5
63.
y
0.1
x
2
0
0
Ϫ4
1
2
Ϫ0.1
Answers to Practice Exercises
1. 2
2. 6p
10.3 Graphs of y = a sin (bx + c) and y = a cos (bx + c)
In the function y = a sin 1bx + c2 , c represents the phase angle. It is another very
important quantity in graphing the sine and cosine functions. Its meaning is illustrated
in the following example.
1IBTF"OHMF t %JTQMBDFNFOU t
Graphs of y = a sin (bx + c) and
y = a cos (bx + c)
E X A M P L E 1 Sketch of a function with phase angle
Sketch the graph of y = sin 1 2x + p4 2 .
Here, c = p>4. Therefore, in order to obtain values for the table, we
assume a value for x, multiply it by 2, add p>4 to this value, and then
find the sine of the result. The values shown are those for which
2x + p>4 = 0, p>4, p, 2, 3p>4, p, and so on, which are the important
values for y = sin 2x.
y
1
p
−8 0
p
4
p
2
3p
4
p
x
x
- p8
0
p
8
p
4
3p
8
p
2
5p
8
3p
4
7p
8
p
y
0
0.7
1
0.7
0
-0.7
-1
-0.7
0
0.7
−1
Fig. 10.14
Solving 2x + p>4 = 0, we get x = -p>8, which gives y = sin 0 = 0. The other
values for y are found in the same way. See Fig. 10.14.
■
x
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304
CHAPTER 10 Graphs of the Trigonometric Functions
Note from Example 1 that the graph of y = sin 1 2x + p4 2 is precisely the same as
the graph of y = sin 2x, except that it is shifted p>8 units to the left. Fig. 10.15 shows
the graphs of y = sin 2x and y = sin 1 2x + p4 2 . We see that the shapes are the same
and that the graph of y = sin 1 2x + p4 2 is about 0.4 unit (p>8 ≈ 0.39) to the left of
the graph of y = sin 2x.
In general, the effect of c in the equation y = sin1bx + c2 can be understood
x
when we write bx + c = b1x + bc 2. This means that the function y = sin1 bx + c2
is obtained as a result of adding the constant c>b to x in the function y = sin bx. As we
discussed in Section 3.5, by adding a constant to x, the graph of y = a sin bx is shifted
to the left or to the right. In this case, the graph is shifted to the left if c 7 0 and to the
right if c 6 0. The direction and magnitude of the shift is called the displacement (or
phase shift), and it is given by –c>b. Note that the displacement can be obtained by
solving for x in the equation bx + c = 0. In Example 1, the displacement is -p>8.
We can verify that the displacement is -c>b by noting corresponding points on the graphs
of y = sin bx and y = sin1bx + c2. For y = sin bx, when x = 0, then y = 0. For
y = sin1bx + c2, when x = -c>b, then y = 0. The point 1 -c>b, 02 on the graph of
y = sin1bx + c2 is -c>b units to the left of the point 10, 02 on the graph of y = sin bx.
We can use the displacement combined with the amplitude and the period along
with the other information from the previous sections to sketch curves of the functions
y = a sin1bx + c2 and y = a cos 1bx + c2, where b 7 0.
y
Period = p
1
0
p
y = sin (2x + 4 )
p
4
3p
4
p
2
5p
4
p
y = sin 2x
−1
p
shift
8
Fig. 10.15
LEARNING TIP
Carefully note the difference
between y = sin1bx + c 2 and
y = sin bx + c. Writing sin1bx + c 2
means to find the sine of the quantity bx + c, whereas sin bx + c means
to find the sine of bx and then add
the value c.
Important Quantities for Sketching Graphs of y = a sin (bx + c) and
y = a cos (bx + c)
Amplitude = 0 a 0
2p
Period =
b
c
Displacement = b
LEARNING TIP
Note that the constant c and the displacement - c>b differ in sign:
t *Gc 7 0, the graph is shifted to the
left, and displacement is negative.
(10.1)
These quantities allow us to evaluate the function at key values each one-fourth
period. Table 10.3 summarizes these key values for the cycle that starts at x = -c>b
and is completed at x = -c>b + period. A general illustration of the graph of
y = a sin1bx + c2 is shown in Fig. 10.16.
t *Gc 6 0, the graph is shifted to the
right, and displacement is positive.
Table 10.3
Key values
(one cycle)
- bc
- bc +
- bc +
- bc +
y = a sin 1bx + c2
0
y = a cos 1 bx + c2
a
0
0
-a
-a
0
0
a
period
4
period
2
3 # period
4
- bc + period
y
a
c
−b
y
2p
b
Since c < 0,
−c/b is positive
c
0
−b +
−a
y = a sin (bx + c), c > 0
Fig. 10.16
a
(a)
2p
b
x
0
For each
a > 0, b > 0
2p
b
a
c
c
−b
−b +
−a
y = a sin (bx + c), c < 0
(b)
2p
b
x