Calculus: Problems and Solutions

In the last post we talked about two important trig identities. I said you should memorize them (and you probably should), but what if you forget? I want to talk about a powerful way to derive trig identities by remembering just a couple of facts. This is great if you’re forgetful like me! To get started, though, we need to look at a wonderful formula discovered by Leonhard Euler. It’s called, appropriately enough, Euler’s Formula:

Here is a thing whose square is negative one: .

If this formula doesn’t seem weird, there’s probably something wrong with you. What the heck does it mean to raise to an imaginary power? That’s bizarre. But it works, as Euler found, and using this formula you can deduce all sorts of interesting things.

One of the most beautiful formulas of mathematics follows from putting in in Euler’s formula. In that case, after adding one to both sides we get

,

a formula that relates five of the most important mathematical constants to one another.