# Complex number de moivre theorem

Here's image and Exercise 3 Hi folks, I would like to ask some explanation of 3rd exercise. To be precise how we get $$4^{31}(\cos \frac{7\pi}{6}+i\sin \frac{7\pi}{6})$$?

Please could you explain it step by step, Yes I know how de Moivre theorem works but I'm confused with this number $$41$$ transforming to $$31$$ Thank you!

• Welcome to MSE. Here is a tutorial on typesetting math on this site. Commented Oct 17, 2019 at 21:13
• The changing of $41$ to $31$ is a typo. It should be $41$.
– MPW
Commented Oct 17, 2019 at 21:21
• @MPW okay if so how we got $4^{31}(\cos \frac{7\pi}{6}+i\sin \frac{7\pi}{6})$? Commented Oct 17, 2019 at 21:24
• See my answer posted below.
– MPW
Commented Oct 17, 2019 at 21:32

Next, the notation is misleading. They mean that $$(41\times\frac{11}{6})\pi = \frac{451}{6}\pi=(74+\frac{7}{6})\pi$$. The idea is to leave an even integer out front, since it will produce a whole multiple of $$2\pi$$.
Then they reduce this to an angle between $$0$$ and $$2\pi$$, which would be $$\frac{7}{6}\pi$$.