Thread: Prooving DeMoivre's Theorem

I am having so much trouble prooving this theorem. This is what I have to prove:

PROVE: If z = r(cos u + i sin u), then z^n = r^n(cos nu + i sin nu) when n = 2

I am so frustrated at this point. I have to list all my steps and no matter how I try to fix it, it never comes out correctly.

Please help.

Originally Posted by missyd819

I am having so much trouble prooving this theorem. This is what I have to prove:

PROVE: If z = r(cos u + i sin u), then z^n = r^n(cos nu + i sin nu) when n = 2

I am so frustrated at this point. I have to list all my steps and no matter how I try to fix it, it never comes out correctly.

Please help.

What you are being asked to prove is:



Well:



but we have identities:



and



Hence:



as required

RonL

In general, it is true for any integer .

note that you can also write as

thus, if we have



then transform the RHS to trigonometric form, you will get

Originally Posted by kalagota

In general, it is true for any integer .

note that you can also write as

thus, if we have



then transform the RHS to trigonometric form, you will get

Once you can write a complex number in exponential form there is nothing left to prove!

RonL