Trigonometry Trigonometric Functions
The Sine Function
   Concepts 1

Definition 1
.
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 AP. 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
.
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
  1. The domain of the sine function is the set of real numbers .
  2. The range of the sine function is interval [-1,1].
  3. The sine repeats its values every units, that is, the sine function has period .
  4. 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.
  5.  The x-intercepts are for every integer n. That is, .

Important Identities
For every real number x


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