<s>[INST] Expressing complex numbers in exponential form I do not know why I am having so many issues with complex numbers. I am basically trying to teach myself, but keep doubting myself and getting very frustrated. I've no way of checking my answers so I'm just lost. So I want to write the following in polar form $re^{i\theta}$ (although I thought that was called exponential form?) and $-\pi < \theta \leq \pi$.
My first question is expressing $(\cos(\frac{2\pi}{9}) + i\sin(\frac{2\pi}{9}))^3$. Using De Moivre, this is $$\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3})$$ so it is $e^{i(\frac{2\pi}{3})}$ yes?
My next one is to express $\frac{2+2i}{-\sqrt{3} +i}$. So my attempt is to multiply by the complex conjugate and we get that this is $\frac{-\sqrt{3} +1}{2} + i\frac{-1 - \sqrt{3}}{2}$. Find $r$ which is the modulus which is $\sqrt2$ and $\theta$ is just something I cannot seem to find. I know if $z = a+ib$ then $\theta = \tan^{-1}(\frac{b}{a})$ but I'm stuck even with the $(\frac{b}{a})$ and am confusing myself of what quadrant to look in.
Lastly, I have no idea how to express the next one which is $\frac{4i}{3e^{(4+i)}}$.
Any help and direction at all is appreciated.
 [/INST] *

*Yes, $\left(\cos\left(\frac{2\pi}9\right)+i\sin\left(\frac{2\pi}9\right)\right)^3=\left(\cos\left(\frac{2\pi}3\right)+i\sin\left(\frac{2\pi}3\right)\right)$.

*Note that$$\frac{2+2i}{-\sqrt3+i}=\frac{1+i}{-\frac{\sqrt3}2+\frac12i}.$$Furthermore, $1+i=\sqrt2\left(\cos\left(\frac{\pi}4\right)+i\sin\left(\frac{\pi}4\right)\right)$ and $-\frac{\sqrt3}2+\frac12i=\cos\left(\frac{5\pi}6\right)+i\sin\left(\frac{5\pi}6\right)$. So, the quotient is equal to$$\sqrt2\left(\cos\left(\frac\pi4-\frac{5\pi}6\right)+i\sin\left(\frac\pi4-\frac{5\pi}6\right)\right)=\sqrt2\left(\cos\left(-\frac{7\pi}{12}\right)+i\sin\left(-\frac{7\pi}{12}\right)\right).$$

*Note that$$\frac{4i}{3e^{4+i}}=\frac43\times\frac{e^{\frac{\pi i}2}}{e^{4+i}}=\frac43e^{-4+\frac{\pi i}2-i}.$$

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