Find *z*, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]

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#### Solution

We know that,

\[z = \left| z \right|\left\{ \cos\left[ \arg(z) \right] + i\sin\left[ \arg(z) \right] \right\}\]

\[ \Rightarrow z = 4\left( \cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6} \right)\]

\[ = 4\left( - \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\]

\[ = 4\left( - \frac{\sqrt{3}}{2} + \frac{1}{2}i \right)\]

\[ = - 2\sqrt{3} + 2i\]

Thus,

\[z = - 2\sqrt{3} + 2i\]

Concept: Concept of Complex Numbers

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