Intersections of Polar Functions Examples

Find the points at which r = 1 and r = sin θ intersect.

We have 1 = sin θ when . This makes sense when we look at a graph of the two functions: 

They intersect at , and we can see from the graph that this is the only intersection point.

Find where r = cos θ and r = sin θ intersect.

Start by graphing the functions:

We can see that they intersect at two places. Set the equations equal:
cos θ = sin θ

This is true when  

Then

.

One of our points of intersection is

.

We can see from the picture that these graphs also intersect at the origin, but we can't find that by solving the equation

cos θ = sin θ because

different values of θ make cos θ and sin θ equal to 0.

This doesn't matter since the point (0, θ) is the same point no matter what θ is.

We may as well call the point r = 0 and not worry about θ. This point, r = 0, is the second intersection point.