# Roots of complex equation

1. Feb 13, 2007

### buzzmath

1. The problem statement, all variables and given/known data
Find all roots of the equation cos(z)=2 (z is a complex number)

2. Relevant equations

3. The attempt at a solution
What do they mean find the roots of this equation? We're just going over trig functions and it doesn't say anything about roots so I'm not sure what they're asking for. when I looked on the internet I just kept getting things raised to 1/n. I don't see that all here.

Last edited: Feb 13, 2007
2. Feb 13, 2007

### Gib Z

Okily well the roots basically means the solutions. We want a z that which, cos z=2 ok?

Write out cos x in terms of the exponential function, (which can be derived from Euler Formula).

$$cos x = \frac{e^{ix} + e^{-ix}}{2}$$. Now write z in terms of its reals and imaginarys, z=a + bi. Substitute and get

$$e^{iz} + e^{-iz} = 4$$. Hopefully you can work it from there.