# How do you convert rectangular coordinates to polar coordinates?

Dec 1, 2014

To convert from polar to rectangular:

$x = r \cos \theta$
$y = r \sin \theta$

To convert from rectangular to polar:

${r}^{2} = {x}^{2} + {y}^{2}$
$\tan \theta = \frac{y}{x}$

This is where these equations come from: Basically, if you are given an $\left(r , \theta\right)$ -a polar coordinate- , you can plug your $r$ and $\theta$ into your equation for x=rcos theta  and $y = r \sin \theta$ to get your $\left(x , y\right)$.

The same holds true for if you are given an $\left(x , y\right)$-a rectangular coordinate- instead. You can solve for $r$ in ${r}^{2} = {x}^{2} + {y}^{2}$ to get $r = \sqrt{{x}^{2} + {y}^{2}}$ and solve for $\theta$ in $\tan \theta = \frac{y}{x}$ to get $\theta = \arctan \left(\frac{y}{x}\right)$ (arctan is just tan inverse, or ${\tan}^{-} 1$). Note that there can be infinitely many polar coordinates that mean the same thing. For example, $\left(5 , \frac{\pi}{3}\right) = \left(5 , - 5 \frac{\pi}{3}\right) = \left(- 5 , 4 \frac{\pi}{3}\right) = \left(- 5 , - 2 \frac{\pi}{3}\right)$...However, by convention, we are always measuring positive $\theta$ COUNTERCLOCKWISE from the x-axis, even if our $r$ is negative.

Let's look at a couple examples.

( 1)Convert $\left(4 , 2 \frac{\pi}{3}\right)$ into Cartesian coordinates.

So we just plug in our $r = 4$ and $\theta = 2 \frac{\pi}{3}$ into

$x = 4 \cos 2 \frac{\pi}{3} = - 2$
$y = 4 \sin 2 \frac{\pi}{3} = 2 \sqrt{3}$

The cartersian coordinate is $\left(- 2 , 2 \sqrt{3}\right)$

(2) Convert $\left(1 , 1\right)$ into polar coordinates. ( since there are many posibilites of this, the restriction here is that $r$ must be positive and $\theta$ must be between 0 and $\pi$)

So, $x = 1$ and $y = 1$. We can find $r$ and $\theta$ from:
$r = \sqrt{{1}^{2} + {1}^{2}} = \sqrt{2}$
$\theta = \arctan \left(\frac{y}{x}\right) = \arctan \left(1\right) = \frac{\pi}{4}$

The polar coordinate is $\left(\sqrt{2} , \frac{\pi}{4}\right)$