Question

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

Answer 1

Given:

`r=2-3cos(theta)" [1]"`
`r=cos(theta)" [2]"`

To prevent the same points on the curve from being repeated with different angles, we restrict the domain for both equations to `0<= theta < 2pi`

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

`2-3cos(theta)=cos(theta),0<= theta < 2pi`

`4cos(theta) = 2,0<= theta < 2pi`

`cos(theta) = 1/2,0<= theta < 2pi`

`theta = pi/3` and `theta = (5pi)/3`

The points in polar coordinates are `(1/2, pi/3) and (1/2,(5pi)/3)`

NOTE: Both curves produce a point where `r = 0` but it is not an intersection point, because the angle for equation [1] is `theta =cos^-1(2/3)` and the angle for equation [2] is `theta = pi/2`