# Trigonometric Functions of a General Angle in .NET Print QR Code JIS X 0510 in .NET Trigonometric Functions of a General Angle

CHAPTER 2 Trigonometric Functions of a General Angle
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Fig. 2.8
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For any angle , cos x/r and sin y/r. On a unit circle, r 1 and the arc length s r and cos cos s x/1 x and sin sin s y/1 y. The point P associated with the arc length s is determined by P(x, y) P(cos s, sin s). The wrapping function W maps real numbers s onto points P of the unit circle denoted by W(s) (cos s, sin s)
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Some arc lengths are paired with points on the unit circle whose coordinates are easily determined. If s 0, the point is (1, 0); for s /2, one-fourth the way around the unit circle, the point is (0, 1); s is paired with ( 1, 0); and s 3 /2 is paired with (0, 1). (See Sec. 1.5.) These values are summarized in the following chart.
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s 0 /2 P(x, y) (1, 0) (0, 1) ( 1, 0) 3 /2 (0, 1) cos s 1 0 1 0 sin s 0 1 0 1
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Circular Functions
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Each arc length s determines a single ordered pair (cos s, sin s) on a unit circle. Both s and cos s are real numbers and define a function (s, cos s) which is called the circular function cosine. Likewise, s and sin s are real numbers and define a function (s, sin s) which is called the circular function sine. These functions are called circular functions since both cos s and sin s are coordinates on a unit circle. The circular functions sin s and cos s are similar to the trigonometric functions sin u and cos u in all regards, since, as shown in Chap. 1, any angle in degree measure can be converted to radian measure, and this radian-measure angle is paired with an arc s on the unit circle. The important distinction for circular functions is that since (s, cos s) and (s, sin s) are ordered pairs of real numbers, all properties and procedures for functions of real numbers apply to circular functions. The remaining circular functions are defined in terms of cos s and sin s. tan s cot s sec s csc s sin s cos s cos s sin s 1 cos s 1 sin s for s 2 p 2 kp where k is an integer
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for s 2 kp where k is an integer for s 2 p 2 kp where k is an integer
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for s 2 kp where k is an integer
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CHAPTER 2 Trigonometric Functions of a General Angle
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It should be noted that the circular functions are defined everywhere that the trigonometric functions are defined, and the values left out of the domains correspond to values where the trigonometric functions are undefined. In any application, there is no need to distinguish between trigonometric functions of angles in radian measure and circular functions of real numbers.
SOLVED PROBLEMS
2.1 Using a rectangular coordinate system, locate the following points and find the value of r for each: A(1, 2), B( 3, 4), C( 3, 3 23), D(4, 5) (see Fig. 2.9).
For A: r For B: r For C: r For D: r 2x2 29 29 216 y2 16 27 25 21 5 6 241 4 25
Fig. 2.9
2.2 Determine the missing coordinate of P in each of the following: (a) (b) (c) (d) x x y x 2, r 3, P in the first quadrant 3, r 5, P in the second quadrant 1, r 3, P in the third quadrant 2, r 25, P in the fourth quadrant (e) (f) (g) (h) x y x y 3, r 3 2, r 2 0, r 2, y positive 0, r 1, x negative
5 and y 25. 4. 2 22. 1. 0. 25.
(a) Using the relation x2 y2 r2, we have 4 y2 9; then y2 Since P is in the first quadrant, the missing coordinate is y (b) Here 9 y2 25, y2 16, and y 4. Since P is in the second quadrant, the missing coordinate is y (c) We have x 1 9, x 8, and x 2 22. Since P is in the third quadrant, the missing coordinate is x
(d) y
4 and y r
1. Since P is in the fourth quadrant, the missing coordinate is y 9 9 0. 0 and the missing coordinate is y