Problem Statement
Find the modulus and principal argument of the complex number $$z = -1 - i\sqrt{3}$$, and express it in polar form.
Verified Solution & Marking Scheme
Calculate Modulus r
$$r = |z| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$$
Find reference angle and quadrant
$$\tan \alpha = \left|\frac{-\sqrt{3}}{-1}\right| = \sqrt{3} \implies \alpha = \frac{\pi}{3}$$. Since both real and imaginary parts are negative, $$z$$ lies in the third quadrant.
Determine Principal Argument
$$\theta = -(\pi - \alpha) = -\left(\pi - \frac{\pi}{3}\right) = -\frac{2\pi}{3}$$
Write in Polar Form
$$z = r(\cos \theta + i\sin \theta) = 2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)$$