Question

Let $z$ be a complex number such that $|z|=4$ and $arg\left(z\right)=\frac{5\pi }{4}$ then find $z$.

Answer 1

Given, $z$ be a complex number such that $|z|=4$ and $arg\left(z\right)=\frac{5\pi }{4}$.

Now $z$ can be written as

$z=|z|e^{i\frac{5\pi }{4}}$

or, $z=4(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4})$

or, $z=4(-\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})$

or, $z=-2\sqrt{2}+i2\sqrt{2}$

or, $z=2\sqrt{2}(-1+i)$.