How do you graph the system of polar equations to solve #r=2+2costheta# and #r=3+sintheta#?

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Answer 1 · 2017-02-05T12:22:16

See the graph for your pair of a cardioid and a limacon. I have used Socratic utility. Common points are #(2, -pi/2) and (3.79, 26.57^o)#.

Here, use cartesian forms of the equations :

For the cardioid, it is from

#r^2=x^2+y^2=2r +2rcostheta=2sqrt(x^2+y^2)+2x#

and, for the limacon, it is from

#r^2=x^2+y^2=3r + rsintheta=3sqrt(x^2+y^2)+y#

It is revealed that one common point is on

#theta = -pi/2#, at which r = 2.

We can measure the other angle or solve

#r=2+2costheta=3+sintheta#. This gives

#cos(theta+cos^(-1)(2/sqrt5))=1/sqrt5#, from which

#cos(theta+26.57^0)=cos(63.43^o)#, and so,

#theta=26.57^o^ and r = 3.79#, nearly.

For the second point, use

#theta+cos^(-1)(2/sqrt5)=2pi+cos^(-1)(1/sqrt5)#, giving

#theta=360^o-(26.57^0+63.43^o)=270^o# that is equivalent to

#-90^o#.

graph{(x^2+y^2-2sqrt(x^2+y^2)-2x)(x^2+y^2-3sqrt(x^2+y^2)-y)=0 [-10, 10, -5, 5]}