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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.

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