Question

Find the quotient of the complex numbers. Leave answer in polar form.
z1=1/8(cos2pi/3 + i sin2pi/3)
z2=1/3(cospi/4 + i sinpi/4)

(answer choices are in the image below)

Answer 1

$\bf \qquad \textit{division of two complex numbers} \\\\ \cfrac{r_1[cos(\alpha)+isin(\alpha)]}{r_2[cos(\beta)+isin(\beta)]}\implies \cfrac{r_1}{r_2}[cos(\alpha - \beta)+isin(\alpha - \beta)]\\\\ -------------------------------$

$\bf \cfrac{z1}{z2}\implies \cfrac{\frac{1}{8}\left[cos\left(\frac{2\pi }{3} \right)+i~sin\left(\frac{2\pi }{3} \right) \right]} {\frac{1}{3}\left[cos\left(\frac{\pi }{4} \right)+i~sin\left(\frac{\pi }{4} \right) \right]} \\\\\\ \cfrac{\quad \frac{1}{8}\quad }{\frac{1}{3}}\left[cos\left(\frac{2\pi }{3}-\frac{\pi }{4} \right)+i~sin\left(\frac{2\pi }{3}-\frac{\pi }{4} \right) \right] \\\\\\ \cfrac{3}{8}\left[cos\left(\frac{5\pi }{12} \right)+i~sin\left(\frac{5\pi }{12} \right) \right]$