# How do you convert  3i to polar form?

Sep 7, 2016

Use $z = r \left(\cos \theta + i \sin \theta\right)$

#### Explanation:

A complex number takes the form $z = a + b i$.
In this example, $a = 0$ and $b = 3$ because $z = 0 + 3 i$.

To find $r$, use the pythagorean theorem.
$r = \sqrt{{a}^{2} + {b}^{2}}$.
$r = \sqrt{{0}^{2} + {3}^{2}} = 3$

To find $\theta$, think about $a$ as a value along the x-axis, and $b$ as a value along the y axis. In this case, $a$ is zero. So, we have a point on the positive y axis, which implies $\theta = \frac{\pi}{2}$.

Substitute r and theta into $z = r \left(\cos \theta + i \sin \theta\right)$
$z = 3 \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$

Of course, $\cos \left(\frac{\pi}{2}\right) = 0$, so some teachers would give the answer as $z = 3 i \sin \left(\frac{\pi}{2}\right)$.