




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 e^{ix} = cosx + isinx.
The the Pythagorean Theorem is the statement e^{ix}e^{ix} = e^{0} = 1;
The periodicity condition follw from the fact that ;
The addition theorems are the statement that e^{i(x+y)} = e^{ix}e^{iy}.
The expansions
also follow immediately as do the relations
The periodicity conditions follow from the fact that ;
The addition theorems are the statement that e^{i(x+y)} = e^{ix}e^{iy}
Since cos has the real (that is even) terms of the series for e^{ix}, we can write
and similarly,