# How do you express z = 1 + i sqrt(3) in polar and trigonometric form, and sketch z on the complex number plane?

Nov 18, 2017

The polar form is $\left(2 , \frac{\pi}{3}\right)$. The trigonometric form is $z = 2 \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right)$

#### Explanation:

The polar form is $r = f \left(\theta\right)$

$z = \cos \theta + i \sin \theta$

The modulus of $z$ is

$| z | = | 1 + \sqrt{3} i | = \sqrt{1 + {\left(\sqrt{3}\right)}^{2}} = \sqrt{4} = 2$

$r = \frac{z}{| z |} = \left(2\right) \left(\frac{1}{2} + \frac{\sqrt{3}}{2} i\right)$

So,

$\cos \theta = \frac{1}{2}$ and $\sin \theta = \frac{\sqrt{3}}{2}$

Therefore,

$\theta = \frac{\pi}{3}$ $\mod 2 \pi$

So,

The polar form is $\left(2 , \frac{\pi}{3}\right)$

The trigonometric form is $z = 2 \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right)$

graph{((x-1)^2+(y-sqrt3)^2-0.001)=0 [-1.313, 4.163, -0.42, 2.316]}