I have found the rcoordinate to be the sqrt13 which is right

I have found the r-coordinate to be the sqrt(13), which is right, but I am having trouble calculating theta. If you could please go through the process of finding the theta coordinate and writing it in simplified form, that would be appreciated. Thank you so much.

Solution

x = r cos theta

y = r sin theta

y/x = (r sin theta)/(r cos theta) = (sin theta)/(cos theta) = tan theta

theta = arctan(y/x)

(x, y) = (2, -3)

theta = arctan(-3/2) -0.983

Since 0 <= theta <= 2pi, theta = arctan(-3/2) + 2pi 5.300

Check:

x = r cos theta = sqrt(13) cos(arctan(-3/2) + 2pi) = 2

y = r sin theta = sqrt(13) sin(arctan(-3/2) + 2pi) = -3