Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use. in .NET framework

Making QR Code in .NET framework Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.

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SINE AND COSINE FUNCTIONS
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[CHAP. 26
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(b) Let = . If we rotate OA through radians in the counterclockwise direction, the nal position B is ( 1, 0) [see Fig. 26-2(b)]. So, cos = 1 sin = 0
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Fig. 26-2
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(c) Let = 3 /2. Then the nal position B, after a rotation through 3 /2 radians in the counterclockwise direction, is (0, 1) [see Fig. 26-2(c)]. Hence, cos 3 =0 2 sin 3 = 1 2
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(d) Let = 0. If OA is rotated through 0 radians, the nal position is still (1, 0). Therefore, cos 0 = 1 sin 0 = 0
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(e) Let be an acute angle (0 < < /2) of right triangle DEF, and let OBG be a similar triangle with hypotenuse 1 (see Fig. 26-3). By proportionality, BG = b/c and OG = a/c. So, by de nition, cos = a/c, sin = b/c. This agrees with the traditional de nitions: opposite side adjacent side cos = sin = hypotenuse hypotenuse
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Fig. 26-3
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SINE AND COSINE FUNCTIONS
Table 26-1 radians 0 /6 /4 /3 /2 3 /2 degrees 0 30 45 60 90 180 270 cos 1 3/2 2/2 1/2 0 1 0 sin 0 1/2 2/2 3/2 1 0 1
Consequently, we can appropriate the values of the functions for = /6, /4, /3 from high-school trigonometry. The results are collected in Table 26-1, which ought to be memorized.
The above de nition implies that the signs of cos and sin are determined by the quadrant in which point B lies. In the rst quadrant, cos > 0 and sin > 0. In the second quadrant, cos < 0 and sin > 0. In the third quadrant, cos < 0 and sin < 0. In the fourth quadrant, cos > 0 and sin < 0 (see Fig. 26-4).
Fig. 26-4
PROPERTIES We list the following theorems, which give the most important properties of the sine and cosine functions.
Theorem 26.1: sin2 + cos2 = 1. Proof: In Fig. 26-1, the length of OB is given by (2.1), 1= cos 0
+ (sin 0)2 =
(cos )2 + (sin )2
Squaring both sides, and using the conventional notations (sin )2 sin2 and (cos )2 cos2 gives Theorem 26.1. Corollary: 1 sin2 = cos2 and 1 cos2 = sin2 .
SINE AND COSINE FUNCTIONS
[CHAP. 26
EXAMPLE Let be the radian measure of an acute angle such that sin = 3 . By Theorem 26.1, 5
3 2 + cos2 = 1 5 9 + cos2 = 1 25 cos2 = 1 16 9 = 25 25 4 16 = cos = 25 5
Since is an acute angle, cos is positive; cos = 4 . 5
We have already seen ( 25) that two angles that differ by a multiple of 2 radians (360 ) have the same sides. This establishes: Theorem 26.2: The cosine and sine functions are periodic, of period 2 ; that is, for all , cos( + 2 ) = cos sin( + 2 ) = sin
(Moreover, 2 is the smallest positive number with this property.) In view of Theorem 26.2, it is suf cient to know the values of cos and sin for 0 < 2 .
7 3 = sin 2 + = sin = 3 3 3 2 (b) cos 5 = cos (3 + 2 ) = cos 3 = cos ( + 2 ) = cos = 1 = cos (30 + 360 ) = cos 30 = 3 (c) cos 390 2 2 (d) sin 405 = sin (45 + 360 ) = sin 45 = 2 (a) sin
EXAMPLES
Theorem 26.3: cos is an even function, and sin is an odd function. The proof is obvious from Fig. 26-5. Points B and B have the same x-coordinates, cos ( ) = cos but their y-coordinates differ in sign, sin ( ) = sin Because of Theorem 26.3, we now need to know the values of cos and sin only for 0 . Consider any point A(x, y) different from the origin O, as in Fig, 26-6. Let r be its distance from the origin and let be the radian measure of the angle that line OA makes with the positive x-axis. We call r and polar coordinates of point A. Theorem 26.4: The polar coordinates of a point and its x- and y-coordinates are related by x = r cos For the proof, see Problem 26.2. y = r sin
CHAP. 26]
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