# How do you find the points of intersection of #r=2-3costheta, r=costheta#?

Given:

Set the right side of equation [1] equal to the right side of equation [2]:

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To find the points of intersection between the polar curves ( r = 2 - 3\cos(\theta) ) and ( r = \cos(\theta) ), we set the two equations equal to each other:

[ 2 - 3\cos(\theta) = \cos(\theta) ]

Solving this equation for ( \theta ), we have:

[ 2 = 4\cos(\theta) ] [ \cos(\theta) = \frac{1}{2} ]

This implies that ( \theta = \frac{\pi}{3} ) or ( \theta = \frac{5\pi}{3} ), as these are the angles where ( \cos(\theta) = \frac{1}{2} ).

Now, to find the corresponding values of ( r ) at these angles:

When ( \theta = \frac{\pi}{3} ): [ r = 2 - 3\cos\left(\frac{\pi}{3}\right) = 2 - 3\left(\frac{1}{2}\right) = 2 - \frac{3}{2} = \frac{1}{2} ]

When ( \theta = \frac{5\pi}{3} ): [ r = 2 - 3\cos\left(\frac{5\pi}{3}\right) = 2 - 3\left(\frac{1}{2}\right) = 2 - \frac{3}{2} = \frac{1}{2} ]

So, the points of intersection are ( \left(\frac{\pi}{3}, \frac{1}{2}\right) ) and ( \left(\frac{5\pi}{3}, \frac{1}{2}\right) ).

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