Trigonometry Trigonometric Functions
The Sine Function Concepts 1

|  |  | Definition 1 |  |
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The circle with center at the origin and radius 1 in a rectangular coordinate system is called the unit circle.

Directed Length of Arc
If A is the fixed point with coordinates (1,0) and P(x,y) is any point determined by proceeding from A along the circle in the counterclockwise direction, then there is a unique nonnegative number s equal to the length of the arc A P. If we allow one or more complete revolutions about the unit circle before stopping at an arbitrary point P , the arc length s can equal any nonnegative number. Similarly, by considering clockwise rotations from A to points P , we can obtain any negative real number for s .

|  |  | Definition 2 |  |
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Let s be any real number, and let P(x,y) be the unique point on the unit circle for which the directed length of an arc that starts at A and ends at P is s . Then by definition sin s= y .

Since r=1, the arc length formula reduces to s = . That is, the real numbers s representing arc length is numericaly equal to the radian measure of . The value of the sine at the real number s is equal to its values at s radians.

s = |
s | = | .
| = | .
| = | .

Table shows a partial numerical representation of sin x

x | 0 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
sin x | 0 |  |  |  | 1 |  |  |  | 0 |  |  |  | -1 |  |  |  | 0

Properties of the Sine Function The domain of the sine function is the set of real numbers . The range of the sine function is interval [-1,1] . The sine repeats its values every units, that is, the sine function has period . The sine function is an odd function, that is, sin(-x) = -sin x . for every real number x. Equivalently, the sine curve is symmetric with respect to the origin. The x-intercepts are for every integer n. That is, .

Important Identities For every real number x

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