De Moivre's theorem and exponential formEdexcel A-Level Further Maths: Mind map

Exam questions on De Moivre's theorem and exponential form

1. Let $z=\cos\frac{\pi}{9}+i\sin\frac{\pi}{9}$.
   Find the exact value of $z^3+\frac{1}{z^3}$.2 marks
2. The complex number $z=2e^{i\pi/6}$ is given.
   Find $z^4$ in the form $x+iy$, giving exact values.2 marks
3. Let $z=\cos\theta+i\sin\theta$, where $\theta$ is real.
   Use de Moivre's theorem to show that $\sin3\theta=3\sin\theta-4\sin^3\theta$.3 marks