# How do you find the points of intersection of theta=pi/4, r=2?

Feb 21, 2018

Their point of intersection is at $x = \sqrt{2}$ and $y = \sqrt{2}$

#### Explanation:

$\theta = \frac{\pi}{4}$ is a ray emerging from the origin with an angle of $\frac{\pi}{4}$ from the positive $x$-axis. (cyan)

$r = 2$ is a circle of radius 2 centered at the origin. (magenta)

The point of intersection in polar coordinates is (unsurprisingly)

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

To get the cartesian coordinates, apply the following

$x = r \cos \theta = 2 \cos \left(\frac{\pi}{2}\right) = 2 \left(\frac{1}{\sqrt{2}}\right) = \sqrt{2}$
$y = r \sin \theta = 2 \sin \left(\frac{\pi}{2}\right) = 2 \left(\frac{1}{\sqrt{2}}\right) = \sqrt{2}$