# How to determine all points of intersection in a polar coordinate

## Main Question or Discussion Point

Is there a way to find all points of intersection in a polar co ordinate graph without the need to draw the graph. i/e USing algebra??? If so, how?

mfb
Mentor
Intersection of what?
If you have equations for whatever should be intersecting, look for points where both equations are satisfied.

Mark44
Mentor
Finding the points of intersection in polar coordinates can be tricky, since the coordinates of a point don't have to be unique, unlike Cartesian coordinates. For this reason, using algebra techniques alone might not give you all intersections.

For example, consider r = cos(θ) and r = sin(θ). Equating the right sides gives sin(θ) = cos(θ), or tan(θ) = 1, so θ = $\pi/4 + n\pi$, with n an integer.

The two graphs also intersect at the origin, which you probably wouldn't know if you didn't graph them. The reason this intersection point doesn't appear from the algebra work above is that each graph "sees" the origin in different coordinates. For r = cos(θ), the point at the origin is (0, $\pi/2$). For r = sin(θ), the points at the origin are (0, 0) and (0, $\pi$).