Question

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

Answer 1

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)`.

Explanation:

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]}