# What is the polar form of (2,3)?

Mar 17, 2016

$\left(\sqrt{13} , {\tan}^{- 1} \left(\frac{3}{2}\right)\right)$

#### Explanation:

To write in polar form, you need to know

1. the distance from the point to the origin
2. the angle the line passing through it and the origin makes with the positive $x$ axis.

To solve 1. we use Pythagoras Theorem

$r = \sqrt{{2}^{2} + {3}^{2}}$

$= \sqrt{13}$

To solve 2. we first find the quadrant that the point lies in.

$x$ is positive and $y$ is positive $\implies$ quadrant I

Then we can find the angle by directly taking inverse tangent of $\frac{y}{x}$. (Note that this is only applicable to quadrant I).

$\theta = {\tan}^{- 1} \left(\frac{3}{2}\right)$

$\approx 0.983$ (in radians)

Therefore, the polar coordinate is $\left(\sqrt{13} , {\tan}^{- 1} \left(\frac{3}{2}\right)\right)$

Note that the answer above is not unique. You can add any integer multiples of $2 \pi$ to $\theta$ to get other representations of the same point.