The Sine Wave...

DEFINITION:

1. Sine of Theta, with Theta in radians, is the y coordinate of the point on the unit circle bisecting the terminal ray of the angle Theta in standard position: vertex of (0,0) and initial ray coexistant with the positive x axis.

   

   So, rolling out the Sine wave, the y coordinate on the unit circle:

2. Sine of Theta, with Theta in radians, is, the value of the imaginary part of
   1. (-1)^(Theta/Pi); Green with Black Cosine.(x coordinate)
   2. (I)^(Theta/2*Pi); Blue with Black Cosine (x coordinate)
   3. (-I)^(-Theta/2*Pi); Not shown.
   4. Shown in Red: (1)^x = 1.