AS June 2023 Paper 1 Q8

8 Abdoallah wants to write the complex number \(-1 + \mathrm{i}\sqrt{3}\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\) where \(r \geqslant 0\) and \(-\pi \lt \theta \leqslant \pi\)

Here is his method:

\[\begin{array}{ll} r = \sqrt{(-1)^2 + \left(\sqrt{3}\right)^2} \qquad\qquad & \tan\theta = \dfrac{\sqrt{3}}{-1} \\[6pt] \phantom{r} = \sqrt{1 + 3} & \Rightarrow \tan\theta = -\sqrt{3} \\[6pt] \phantom{r} = \sqrt{4} & \Rightarrow \theta = \tan^{-1}\left(-\sqrt{3}\right) \\[6pt] \phantom{r} = 2 & \Rightarrow \theta = -\dfrac{\pi}{3} \end{array}\]\[-1 + \mathrm{i}\sqrt{3} = 2\left(\cos\left(-\frac{\pi}{3}\right) + \mathrm{i}\sin\left(-\frac{\pi}{3}\right)\right)\]

There is an error in Abdoallah’s method.