Graphs and Derivatives of Sine and Cosine Functions in .NET framework

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Graphs and Derivatives of Sine and Cosine Functions
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27.1 GRAPHS Let us rst observe that cos x and sin x are continuous functions. This means that, for any xed x = ,
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lim cos ( + h) = cos
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lim sin ( + h) = sin
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as is obvious from Fig. 27-1. Indeed, as h approaches 0, point C approaches point B. Therefore, the x-coordinate (the cosine) of C approaches the x-coordinate of B, and the y-coordinate (the sine) of C approaches the y-coordinate of B.
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Fig. 27-1
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GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS
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[CHAP. 27
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Table 27-1 x 0 6 4 3 2 2 3 3 4 5 6 cos x 1 sin x 0 1 2
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Now we can sketch the graphs of y = cos x and y = sin x. Table 27-1 contains the values of cos x and sin x for the standard values of x between 0 and /2; these values are taken from Table 26-1. Also listed are the values for 2 /3 (120 ), 3 /4 (135 ), and 5 /6 (150 ). These are obtained from the formulas (see Problem 26.15) = sin and sin + = cos cos + 2 2 The graph of y = cos x is sketched in Fig. 27-2(a). For arguments between and 0, we have used the identity cos ( x) = cos x (Theorem 26.3). Outside the interval [ , ], the curve repeats itself in accordance with Theorem 26.2. The graph of y = sin x [see Fig. 27-2(b)] is obtained in the same way. Notice that this graph is the result of moving the graph of y = cos x to the right by /2 units. This can be veri ed by observing that cos x 2 = sin x
The graphs of y = cos x and y = sin x have the shape of repeated waves, with each complete wave extending over an interval of length 2 (the period). The length and height of the waves can be changed by multiplying the argument and the functional value, respectively, by constants. EXAMPLES
(a) y = cos 3x. The graph of this function is sketched in Fig. 27-3. Because cos 3 x + 2 3 = cos (3x + 2 ) = cos 3x
the function is of period p = 2 /3. Hence, the length of each wave is 2 /3. The number of waves over an interval of length 2 (corresponding to one complete revolution of the ray determining angle x) is 3. This number is called the frequency f of cos 3x. In general, pf = (length of each wave) (number of waves in an interval of length 2 ) = 2
CHAP. 27]
GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS
Fig. 27-2
Fig. 27-3
and so f = 2 p
For an arbitrary constant b > 0, the function cos bx (or sin bx) has frequency b and period 2 /b.
GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS
[CHAP. 27
(b) y = 2 sin x. The graph of this function (see Fig. 27-4) is obtained from the graph of y = sin x (see Fig. 27-2) by multiplying each ordinate by 2. The period (wavelength) and the frequency of the function 2 sin x are the same as those of the function sin x: p = 2 , f = 1. But the amplitude, the maximum height of each wave, of 2 sin x is 2, or twice the amplitude of sin x. note The total oscillation of a sine or cosine function, which is the vertical distance from crest to trough, is twice the amplitude.
Fig. 27-4
For an arbitrary constant A, the function A sin x (or A cos x) has amplitude |A|. (c) Putting together examples (a) and (b) above, we see that the functions A sin bx and A cos bx (b > 0) have period 2 /b, frequency b, and amplitude |A|. Figure 27-5 gives the graph of y = 1.5 sin 4x.
Fig. 27-5
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