Q.Write $z = -\sqrt{3} + i$ in modulus-amplitude form.

Find the modulus $r=|z|$ and the argument $\theta$ (measured from the positive real axis), then write $z=r(\cos\theta+i\sin\theta)$.

Here $z = -\sqrt3 + i$, so $x=-\sqrt3$, $y=1$.

Modulus:

$r = \sqrt{x^2+y^2} = \sqrt{(-\sqrt3)^2+1^2} = \sqrt{3+1} = \sqrt4 = 2$

Argument: since $x<0$ and $y>0$, the point lies in the second quadrant. The reference (acute) angle $\alpha$ satisfies

$\tan\alpha = \left|\frac{y}{x}\right| = \frac{1}{\sqrt3} \implies \alpha = \frac{\pi}{6}$

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