Question

If $z=r(\cos \theta +i\sin \theta )$is a complex number, then

$z^n=\_\_\_\_\_\left[\cos \right(\_\_\_\_)+i\sin (\_\_\_\_\left)\right]$.

Short answer

The general solution is

$z^n=r^n\left[\cos \right(n\theta )+i\sin (n\theta \left)\right]$

Step-by-step solution

Step 1. Given

Given complex number in polar form

$z=r\cos \theta +i\sin \theta$

Calculation

$$\begin{gathered} Z_1{Z}_2=r_1(\cos {\theta }_1+i\sin {\theta }_1)\times r_2(\cos {\theta }_2+i\sin {\theta }_2) \\ =r_1{r}_2(\cos {\theta }_1\cos {\theta }_2+i\cos {\theta }_1\sin {\theta }_2+i\sin {\theta }_1\cos {\theta }_2+i^2\sin {\theta }_1\sin {\theta }_2) \\ =r_1{r}_2\left[\right(\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2)] \\ Similaryforntimesageneralisesolutionwillbe \\ {z}^n=r^n[\cos \left(n\theta \right)+i\sin \left(n\theta \right)] \end{gathered}$$