# How do you convert the Cartesian coordinates (0,-10) to polar coordinates?

Sep 12, 2015

$\left(r , \theta\right) = \left(10 , \frac{3 \pi}{2}\right)$

#### Explanation:

When converting from Cartesian to polar coordinates we use the following relationships

$x = r \cos \theta$

$y = r \sin \theta$

$r = \sqrt{{x}^{2} + {y}^{2}}$

We are given the point $\left(0 , - 10\right)$

Proceed by finding $r$. Since $r$ is a length we only want the positive square root.

$r = \sqrt{{\left(0\right)}^{2} + {\left(- 10\right)}^{2}}$

$r = \sqrt{100} = 10$

Now we utilize the other relationships

$x = r \cos \theta$ so we can write

$0 = 10 \cos \theta$

So $\cos \theta = 0$

We know that we are on the y axis at $x = 0$ and $y = - 10$ so our angle is

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

Also

$y = r \sin \theta$ so we can write

$- 10 = 10 \sin \theta$

$\sin \theta = - 1$

This occurs when $\theta = \frac{3 \pi}{2}$

Therefore $\left(r , \theta\right) = \left(10 , \frac{3 \pi}{2}\right)$

Make sure your calculator is in radians