How do you graph the system of polar equations to solve r=2+2cosθ and r=3+sinθ?

1 Answer
A. S. Adikesavan
Feb 5, 2017

See the graph for your pair of a cardioid and a limacon. I have used Socratic utility. Common points are (2,−π2)and(3.79,26.57o).

Explanation:

Here, use cartesian forms of the equations :

For the cardioid, it is from

r2=x2+y2=2r+2rcosθ=2√x2+y2+2x

and, for the limacon, it is from

r2=x2+y2=3r+rsinθ=3√x2+y2+y

It is revealed that one common point is on

θ=−π2, at which r = 2.

We can measure the other angle or solve

r=2+2cosθ=3+sinθ. This gives

cos(θ+cos−1(2√5))=1√5, from which

cos(θ+26.570)=cos(63.43o), and so,

θ=26.57o^andr=3.79, nearly.

For the second point, use

θ+cos−1(2√5)=2π+cos−1(1√5), giving

θ=360o−(26.570+63.43o)=270o that is equivalent to

−90o.

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]}

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