Finding the Power of a Complex Number using DeMoivre's Theorem

Question

The complex number given is $(1+i)^5$ .

I used DeMoivre's Theorem which $Z^n$ = r[(cos $n$$\theta$ + $i$sin $n$$\theta$)]. When solved I got $Z^n$= cos5 + isin5 . However, the answer key says the answer is -4-4$i$ . I have no idea where they got the four from . Am I missing a step ?

Answer 1

Note that

$$(1+i)=\sqrt 2(\cos \pi/4+i\sin \pi /4)$$

then

$$(1+i)^5=(\sqrt 2)^5(\cos 5\pi/4+i\sin 5\pi /4)$$

then convert it again.

Answer 2

Hint: $\require{cancel}\,(1+i)^2=\bcancel{1}+2i+\bcancel{i^2}=2i$, then $(1+i)^5=(1+i)\cdot(1+i)^4=(1+i)\cdot(2i)^2=\ldots$