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!
First, the 41 to 31 business is a typo. It should be 41.
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 $\frac{7}{6}\pi$.