Find z, if |z|=4 and arg(z)=5 6 . - Vidyadip

Question

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

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Answer

We know that, $\text{z}=|\text{z}|\big\{\cos[\text{arg(z)]}+\text{i}\sin[\text{arg(z)}]\big\}$ $\Rightarrow\text{z}=4\Big(\cos\frac{5\pi}{6}+\text{i}\sin\frac{5\pi}{6}\Big)$ $=4\Big(-\cos\frac{\pi}{6}+\text{i}\sin\frac{\pi}{6}\Big)$ $=4\Big(-\frac{\sqrt{3}}{2}+\frac{1}{2}\text{i}\Big)$ $=-2\sqrt{3}+2\text{i}$ Thus, $\text{z}=-2\sqrt{3}+2\text{i}.$

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