"Now if you have two angles, theta and phi, and you write e to the i phi which
is cosine phi plus i sine phi and multiply them you get something with a real
and an imaginary part, the real part is cosine theta cosine phi minus sine
theta times sine phi, remember i times i is minus i, as minus 1, so if you
multiply these two, what you get is - well, let's simplify a bit - cosine of
theta plus phi - that's a high school trigonometry formula.

The cosine of a sum of two angles, cosine cosine minus sine times sine, and
then, plus the imaginary part, is cosine theta sine phi plus cosine phi times
sine theta, and that's just sine of theta plus phi, and by definition, if e to
the i theta is cosine plus i sine, then this is just e to the i theta plus
phi. So from elementary trigonometry we discover that this combination cosine
theta plus i sine theta has all of the properties of an exponential of i
theta.

Now all of trigonometry - anything you ever wanted to remember about
trigonometry - is all stored in this formula. For example, if you couldn't
remember what a cosine of the sum of two angles is, all you need to do is
multiply e to the i theta times e to the i phi in this form, and you'll
discover that cosine theta plus phi is cosine theta cosine phi minus sine
theta sine phi.

So all of trigonometry - I don't know why they take a year to teach
trigonometry in high school - and then at the end of it they come here to
Stanford and they don't know this formula."

-Leonard Susskind