1. The definitions here are exactly equivalent to the usual ones, that sinx is opposite over hypotenuse for a right triangle, viewed from acute angle x.
2. We measure angles in radians, for which x is the distance around the unit circle subtended by an angle x.
3. All the properties of sines and cosines easily follow from the representation eix = cosx + isinx.
The the Pythagorean Theorem is the statement eixe-ix = e0 = 1;
The periodicity condition follw from the fact that ;
The addition theorems are the statement that ei(x+y) = eixeiy.
also follow immediately as do the relations
The periodicity conditions follow from the fact that ;
The addition theorems are the statement that ei(x+y) = eixeiy
Since cos has the real (that is even) terms of the series for eix, we can write