Obtain $z_1,z_2$ and $z_1/z_2$

Question

Given $j = \sqrt{-1}$

$z_1 = 5 \left(\dfrac{\cos 126^\circ{} + j \sin 126^\circ{}}{\cos 72^\circ{} + j \sin 72^\circ{}}\right);$

$z_2 = 2\cos 30^\circ{} + j \sin 30^\circ{};$

Find using algebraic calculations $z_1\cdot z_2$ and $\frac{z_1}{z_2}$.

It is advised to use the exponential form to have fewer calculations. The problem is that $2\cos(30^\circ{})$. It is not a mistake in fact the $r_2$ is given in the results as $\left(\frac{\sqrt {13}}{2}\right)$ so the $2$ is actually only multiplying $\cos x.$

Answer 1

Hint ($i=\sqrt {-1}$): $$ z_1=5\frac{e^{\frac{7\pi} {10}i}}{e^{\frac{4\pi} {10}i}}=5e^{\frac{3\pi} {10}i};\quad z_2=r_2e^{i\phi_2}, $$ where $$ r_2=\frac {\sqrt {13}}2;\quad\phi_2=\arctan\frac1 {2\sqrt3}. $$

Explaination: the general transformation of complex numbers to polar representation reads: $$ x+iy=\operatorname{sign}(x)\sqrt{x^2+y^2}e^{i\arctan\frac yx} $$