de moivre's theorem

[tsh44]

Hello, I need help solving this problem:

Lat z= cos theta + i sin theta

Evaluate the absolute value of z^100.

I think I would have to change this to polar form but I am not sure how to since there are no numbers. I would appreciate any help. Thanks.

 

[pka]

|z|=1
So |z<SUP>100</SUP>|=|z|<SUP>100</SUP>=1.

 

[tsh44]

How did you get the absolute value of z to equal 1?

 

[pka]

\(\displaystyle \L
\begin{array}{l}
z = a + bi\quad \Rightarrow \quad |z| = \sqrt {a^2 + b^2 } \\
z = \cos (\theta ) + i\sin (\theta )\quad \Rightarrow \quad |z| = \sqrt {\cos ^2 (\theta ) + \sin ^2 (\theta )} \\
\end{array}\)

 

[tsh44]

thank you