Question

Write −1 + i $\sqrt{3}$ in polar form

Answer 1

$$\begin{gathered} \mathrm{Let}z=-1+\sqrt{3}i.Then, \\ r=\left|z\right|=\sqrt{{\left[-1\right]}^2+{\left[\sqrt{3}\right]}^2}=2 \\ \mathrm{Let}\tan \alpha =\left|\frac{\mathrm{Im}\left(z\right)}{\mathrm{Re}\left(z\right)}\right| \\ =\sqrt{3} \\ \Rightarrow \alpha =\frac{\mathrm{\pi }}{3} \\ \mathrm{Since}\mathrm{the}\mathrm{point}\mathrm{representing}z\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{second}\mathrm{quadrant}.\mathrm{Therefore},\mathrm{the}\mathrm{argument}\mathrm{of}z\mathrm{is}\mathrm{given}\mathrm{by} \\ \theta =\mathrm{\pi }-\mathrm{\alpha } \\ =\mathrm{\pi }-\frac{\mathrm{\pi }}{3} \\ =\frac{2\mathrm{\pi }}{3} \\ \mathrm{So},\mathrm{the}\mathrm{polar}\mathrm{form}\mathrm{is}r\left(\cos \theta +i\sin \theta \right) \\ \therefore z=2\left(\cos \frac{2\mathrm{\pi }}{3}+i\sin \frac{2\mathrm{\pi }}{3}\right) \end{gathered}$$