Q.Write $z = -\sqrt{7} + i\sqrt{21}$ in the polar form.

Find the modulus $r=|z|$ and argument $\theta$ from $\cos\theta=x/r,\ \sin\theta=y/r$, then write $z=r(\cos\theta+i\sin\theta)$.

Given $z = -\sqrt7 + i\sqrt{21}$, so $x=-\sqrt7$, $y=\sqrt{21}$.

Modulus:

$r = \sqrt{x^2+y^2} = \sqrt{7+21} = \sqrt{28} = 2\sqrt7.$

Argument: since $x<0$ and $y>0$, $z$ lies in the second quadrant.

$\cos\theta = \frac{x}{r} = \frac{-\sqrt7}{2\sqrt7} = -\frac12, \qquad \sin\theta = \frac{y}{r} = \frac{\sqrt{21}}{2\sqrt7} = \frac{\sqrt3}{2}.$

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