# DeMoivre's Theorem

Tags:
1. Mar 14, 2017

### Faiq

1. The problem statement, all variables and given/known data
Can somebody explain to me how the first two steps are performed?

3. The attempt at a solution
I have no idea how to start the question. I tried using an equation for sin^6 x derived by (cos x + i sinx)^6 = cos 6x+isin 6x but the solution becomes way too hard.

#### Attached Files:

• ###### De Moirve's Theroem.jpg
File size:
24.9 KB
Views:
30
2. Mar 14, 2017

### pasmith

If $z = e^{i\theta}$ then $$\cos n\theta = \frac{z^n + z^{-n}}{2} \\ \sin n\theta = \frac{z^n - z^{-n}}{2i}.$$ Hence $$\sin^6 \theta = \frac{(z - z^{-1})^6}{(2i)^6} = -\frac{(z - z^{-1})^6}{64}.$$

3. Mar 14, 2017

### Faiq

Are these equation correct?
$$\sin^n \theta = \frac{(z-z^{-1})^n}{(2i)^n}$$ and $$\cos^n \theta = \frac{(z+z^{-1})^n}{2^n}$$

Last edited: Mar 14, 2017
4. Mar 14, 2017

Yes.