Question

What is the polar form of -3+ v3i?

Answer 1

• z=-3+√3i
• a=-3,b=√3

So

$\\ \rm\Rrightarrow r=\sqrt{a^2+b^2}$

$\\ \rm\Rrightarrow r=\sqrt{(-3)^2+(\sqrt{3})^2}$

$\\ \rm\Rrightarrow r=\sqrt{9+3}$

$\\ \rm\Rrightarrow r=\sqrt{12}$

$\\ \rm\Rrightarrow r=2\sqrt{3}$

Find theta

$\\ \rm\Rrightarrow \theta=tan^{-1}(\dfrac{y}{x})$

$\\ \rm\Rrightarrow \theta=tan^{-1}(\dfrac{\sqrt{3}}{3})$

$\\ \rm\Rrightarrow \theta=tan^{-1}(\dfrac{-1}{\sqrt{3}})$

$\\ \rm\Rrightarrow \theta=\dfrac{5\pi}{6}$

Polar form

$\\ \rm\Rrightarrow r(cos\theta+isin\theta)$

$\\ \rm\Rrightarrow 2\sqrt{3}(cos\dfrac{5\pi}{6}+isin\dfrac{5\pi}{6})$