Sine & Cosine Functions

Draw any angle in standard position.  Also, on the Cartesian rectangular system, draw a circle centered at the origin.  The point where the terminal side of the angle intersects the circle is (x,y).
The part of the terminal side that is the radius of the circle is called r.

x2 + y2 = r2   is the equation of the circle.
r > 0

Definitions:

sin q = y/r            tan q   = y/x            csc q  = r/y

cos q = x/r            cot q  = x/y             sec q  = r/x

Examples:

  1. Find sin q and cos q when the terminal ray of q passes through (5,-12).
    1. x = 5 and y = -12
    2. 25 + 144 = r2

      3.         169 = r2
                13 = r
    3. sin q = y/r = -12/13

      4. cos q = x/r = 5/13
  2. Find sin q if q is in quadrant IV and cos q = 4/5.
    1. x = 4, r = 5
    2. 16 + y2 = 25

      3.         y2 = 9
              y  = -3 (because y is negative in quadrant IV)
    3. sin q = -3/5
Unit Circle

The unit circle is the circle whose radius is 1.  Its equation is x2 + y2 = 1.

Since sin q = y/r and cos q = x/r , then  in the unit circle sin q = y and cos q = x.
  (Also, tan q  = y/x,  csc q  = 1/y,  cot q  = y/x,   sec q  = 1/x)

Thus, by substitution into the equation of the unit circle, cos2q + sin2q = 1.

Since, in the unit circle, -1 < x < 1 and -1 < y < 1, then -1 < cos q < 1 and -1 < sin q < 1.

By virtue of the definitions, if two angles are coterminal, their cos and sin values are equal.
 

Periodicity

sin (q + 360o) = sin q
cos (q + 360o) = cos q

sin (q + 2p) = sin q
cos (q + 2p) = cos q
 

Even and Odd

Even:  cos (-q) = cos q
          sec (-q) = sec q

Odd:   sin (-q) = - sin q
       csc (-q) = - csc q
       tan (-q) = - tan q
       cot (-q) = - cot q

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