Mathematics Extension 2

By first writing $\sqrt{3}+i$ and $\sqrt{3}-i$ in exponential form, express

in the form of $a^b$, where $a$ and $b$ are integers.

Let $\alpha$ be a real number and suppose that $z$ is a complex number such that

You may assume that

for all positive integer $n$. Let

Hence or otherwise, find all solutions of

Find the cube roots of $2-2i$. Give your answer in exponential form.

By considering the roots of $z^3=4\sqrt{2}(1+i)$, show that

Write the complex number $-\sqrt3+i$ in exponential form.

Using Euler’s theorem, show that if $z=e^{i\theta}=\cos\theta+i\sin\theta$, then
$z^n+z^{-n}=2\cos n\theta.$

Hence show that
$\cos^7\theta=\frac1{64}\left(\cos7\theta+7\cos5\theta+21\cos3\theta+35\cos\theta\right).$

Prove that
$\frac{\cos\theta+i\sin\theta-1}{\cos\theta+i\sin\theta+1}=i\tan\frac\theta2.$

The complex numbers $z=2e^{i\pi/2}$ and $w=6e^{i\pi/6}$ are given. Find $zw$, giving the answer in the form $re^{i\theta}$.

By using part (i), or otherwise, show that
$\operatorname{Re}\left(e^{i\pi/10}\right)=\sqrt{\frac{5+\sqrt5}{8}}.$

By expanding $(e^{i\theta}+e^{-i\theta})^4$, show that
$\cos^4\theta=\frac18\bigl(\cos(4\theta)+4\cos(2\theta)+3\bigr).$

Show that for any integer $n$,
$e^{in\theta}+e^{-in\theta}=2\cos(n\theta).$