Question

TRIGONOMETRY - Given the complex numbers ="z_{1} =20 cis 226" alt="z_{1} =20 cis 226" align="absmiddle" class="latex-formula"> and $z_{2} =5 cis 76$, find the quotient $\frac{z_{1} }{z_{2}}$.

Answer 1

This is easily done after writing both complex numbers in exponential form:

$z_1=20\,\mathrm{cis}226^\circ=20e^{226^\circ i}$

$z_2=5\,\mathrm{cis}76^\circ=5e^{76^\circ i}$

Then

$\dfrac{z_1}{z_2}=\dfrac{20e^{226^\circ i}}{5e^{76^\circ i}}=4e^{(226-76)^\circ i}=4e^{150^\circ i}$

In trigonometric form, this is

$\dfrac{z_1}{z_2}=4\,\mathrm{cis}150^\circ=4(\cos150^\circ+i\sin150^\circ)=\boxed{-\dfrac{\sqrt3}2+\dfrac i2}$