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CHAPTER 7 Graphs of the Trigonometric Functions
Fig. 7.3
has amplitude a and period 2 /b. Thus the graph of y 3 sin 2x has amplitude 3 and period 2 /2 . Figure 7.4 exhibits the graphs of y sin x and y 3 sin 2x on the same axes. More complicated forms of wave motions are obtained by combining two or more sine curves. The method of adding corresponding ordinates is illustrated in the following example.
EXAMPLE 7.1 Construct the graph of y
sin x
3 sin 2x. See Fig. 7.4.
First the graphs of y1 sin x and y2 3 sin 2x are constructed on the same axes. Then, corresponding to each x value, we find the y value by finding y1 value for that x, the y2 values for that x, and adding the two values together. For example, when x OA1, y1 A1 B1 and y2 A1 C1, so y A1 B1 A1 C1 A1(B1 C1) A1 D1, when x OA2, y1 A2 B2 and y2 A2 C2, so y A2B2 A2C2 A2(B2 C2) A2 D2, and when x OA3, y1 A3 B3 and y2 A3 C3, so y A3 B3 A3(B3 C3) A3 D3. Thus, for y sin x 3sin 2x, the y-coordinate for any x-coordinate is the A3 C3 algebraic sum of the y-coordinates of y1 sin x and y2 3sin 2x.
CHAPTER 7 Graphs of the Trigonometric Functions
Fig. 7.4
SOLVED PROBLEMS
7.1 Sketch the graphs of the following for one period. (a) y (b) y 4 sin x sin 3x (c) y (d) y 3 sin 2 x 2 cos x 2 sin A x
1 2 1
(e) y pB
3 cos 2 x
1 3 sin A 2 x
In each case we use the same curve, and then put in the y axis and choose the units on each axis to satisfy the requirements of amplitude and period of each curve (see Fig. 7.5).
Fig. 7.5
CHAPTER 7 Graphs of the Trigonometric Functions
(a) y (b) y (c) y (d) y (e) y 4 sin x has amplitude sin 3x has amplitude 3 sin 2 x has amplitude 2 cos x has amplitude 3 cos 2 x has amplitude
4 and period 1 and period 3 and period 2 and period 3 and period
2 . 2 /3. 2 /2 4 .
2 . Note the position of the y axis.
7.2 Construct the graph of each of the following. (a) y
tan x,
(b) y
3 tan x,
(c) y
tan 3x,
(d) y
tan 4x
In each case, we use the same curve and then put in the y axis and choose the units on the x axis to satisfy the period of the curve (see Fig. 7.6).
(a) y
tan x has period
(b) y
3 tan x has period
(c) y
tan 3x has period /3
(d) y
1 tan 4x has period p/ 4
Fig. 7.6
7.3 Construct the graph of each of the following (see Fig. 7.7). (a) y (b) y sin x cos x sin 2x cos 3x (c) y (d) y sin 2x cos 3x 3 sin 2x 2 cos 3x
7.4 Construct a graph of each of the following (see Fig. 7.8). (a) y (b) y 3 sin x 1 sin x 2 (c) y (d) y cos x 2 1 1 2 cos x
CHAPTER 7 Graphs of the Trigonometric Functions
Fig. 7.7
(a) y
3 sin x is shifted right 1 unit
(b) y
sin x is shifted down 2 units
(c) y
cos x is shifted up 2 units
(d) y
cos x is shifted down 1 unit
Fig. 7.8
CHAPTER 7 Graphs of the Trigonometric Functions
7.5 Construct a graph of each of the following (see Fig. 7.9). (a) y (b) y
(a) y
sin(x sin (x
p>6) p>6)
(c) y (d) y
cos (x cos (x
p>4) p>3)
(b) y sin x is shifted left /6 units
sin x is shifted right /6 units
(c) y
cos x is shifted right /4 units
(d) y
cos x is shifted left /3 units
Fig. 7.9
SUPPLEMENTARY PROBLEMS
7.6 Sketch the graph of each of the following for one period (see Fig. 7.10). (a) y (g) y Ans. 3 sin x, (b) y tan 2x (a) y 3 sin x sin 2x, (c) y 4 sin x/2, (d) y 4 cos x, (e) y (b) y 2 cos x/3, (f) y sin 2x 2 tan x,
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