Euler's relation and polar formEdexcel International A Level Further Maths: Mind map

Exam questions on Euler's relation and polar form

1. Let $z=2e^{i\pi/3}$ and $w=3e^{i\pi/6}$.
   Find $z+w$ in the form $a+ib$, giving $a$ and $b$ as exact values.2 marks
2. Let $\theta$ be a real number and let $z=e^{i\theta}$.
   By expanding $\left(z+\dfrac1z\right)^2$ and using $z^2=e^{2i\theta}$, show that $\cos^2\theta=\frac12(1+\cos2\theta)$.2 marks
3. The complex number $z=-1+i\sqrt3$.
   Write $z$ in the form $re^{i\theta}$, where $-\pi<\theta\le\pi$.3 marks