DeMoivre's Theorem, I really don't understand it!

[morganna]

Okay I am taking an online class and I really don't understand the assignment. I don't understand DeMoivre's Theorem. Here's the question:

DeMoivre’s Theorem states, “If z = r(cos u + i sin u), then zn = rn(cos nu + i sin nu)."

1. Verify DeMoivre's Theoremm for n=2. Show all steps in your verification.

I know I have to use the sum of angles identities. I am just very confused about this problem. I would really appreiciate any help anyone can give me.

 

[galactus]

You can find the product of two complex numbers by multiplying their moduli and adding their arguments.

The moduli, \(\displaystyle r = \sqrt{a^{2}+b^{2}}\) is just the Pythagoras thing and the argument is \(\displaystyle \L\\{\theta}=tan^{-1}(b/a)\)

There's a theorem that says,

\(\displaystyle \L\\z_{1}z_{2}=r_{1}r_{2}[cos({\theta}_{1}+{\theta}_{2})+isin({\theta}_{1}+{\theta}_{2})]\)

See?. We multiply the moduli, r, and add the arg(theta).

Now, using that fact, we can let \(\displaystyle z_{1} \;\ and \;\ z_{2}\) both equal \(\displaystyle \L\\r(cos{\theta}+isin{\theta})\)

Then we have \(\displaystyle \L\\z^{2}=r\cdot{r}[cos({\theta}+{\theta})+isin({\theta}+{\theta})]=r^{2}(cos(2{\theta})+isin(2{\theta}))\)

You can apply the same reasoning to find z^3, z^4, .........