Question

Find z, if $\left|z\right|=4\mathrm{and}\arg \left(z\right)=\frac{5\mathrm{\pi }}{6}.$

Answer 1

We know that,

$$\begin{gathered} z=\left|z\right|\left\{\cos \left[\arg \left(z\right)\right]+i\sin \left[\arg \left(z\right)\right]\right\} \\ \Rightarrow z=4\left(\cos \frac{5\mathrm{\pi }}{6}+i\sin \frac{5\mathrm{\pi }}{6}\right) \\ =4\left(-\cos \frac{\mathrm{\pi }}{6}+i\sin \frac{\mathrm{\pi }}{6}\right) \\ =4\left(-\frac{\sqrt{3}}{2}+\frac{1}{2}i\right) \\ =-2\sqrt{3}+2i \end{gathered}$$

Thus, $z=-2\sqrt{3}+2i$.