Separating e^(xi) to form a-bi

[thunderjolt]

Homework Statement

I am in dif eq, but just need to know how to separate a power.

separate e^(xi) into the form a-bi, where x is a constant (in my homework, x is 4pi/3, but that's not too relevant)

i is the imaginary number sqrt(-1)

Homework Equations

I don't know if there is some simple rule, or if I actually need to use calculus and integrals.
The only thing I know is that e^(x+y) = e^(x)e^(y). However, I can't use that here, because the power is the multiple.

The Attempt at a Solution

I tried setting it equal to y = e^(xi) and taking the natural log of both sides, but it just got really messy and I ended up with a square root of i, which is not good.

 

[jbunniii]

Are you looking for the Euler identity?
$$e^{ix} = \cos(x) + i\sin(x)$$

 

[bossman27]

Are you familiar with Taylor series? As jbunniii mentioned, it is the Euler formula, but the easiest way to derive it is by using the Taylor series expansion of e^x, with x = ix, and then separate the real and imaginary terms into two series which are known to be the Taylor series for cosine and sine, respectively.

 

[thunderjolt]

ok, that makes sense, the prof did the taylor series in class...thanks