Question

For the complex number z=5sqrt3/4 - 5/4i, what is the polar form?

Answer 1

Answer:

$\displaystyle z=\frac{5}{2}\biggr(\cos\frac{11\pi}{6}+i\sin\frac{11\pi}{6}\biggr)$

Step-by-step explanation:

A complex number in polar form is $z=r(\cos\theta+i\sin\theta)$ where $r=\sqrt{x^2+y^2}$ and $\displaystyle \theta=\tan^{-1}\biggr(\frac{y}{x}\biggr)$, thus:

$r=\sqrt{x^2+y^2}=\sqrt{(\frac{5\sqrt{3}}{4})^2+(-\frac{5}{4})^2}=\sqrt{\frac{75}{16}+\frac{25}{16}}=\sqrt{\frac{100}{16}}=\frac{10}{4}=\frac{5}{2}$

$\displaystyle \theta=\tan^{-1}\biggr(\frac{y}{x}\biggr)=\tan^{-1}\biggr(\frac{-\frac{5}{4}}{\frac{5\sqrt{3}}{4}}\biggr)=\tan^{-1}\biggr(-\frac{\sqrt{3}}{3}\biggr)=\frac{11\pi}{6}$

Thus, the complex number in polar form is $\displaystyle z=\frac{5}{2}\biggr(\cos\frac{11\pi}{6}+i\sin\frac{11\pi}{6}\biggr)$