# Proof 5.25

Use algebra and a graph to find all points where the curves  and  intersect.
Proof: Consider the polar equations  and  as given. Since both are set equal to r, we can set them equal to one another such that . From this, we can further our algebra as follows:  since each side shares a common 3  by trig double angle identity of   by factoring  and                                               In the sketch above, we can see that there are three intersection points of the polar equations. Through algebra, we were also able to show that there are three intersection points of the polar equations. Point A is , Point B is  instead of because inverse sine is restricted and Point C is . We are able to verify these points if we start at the horizontal axis, a,  and rotate counterclockwise. The point D may appear to be an intersection point at . However if we trace the functions, we see that the the functions do not intersect at that point because they pass through it at a different theta value.