Question

How do you find the points of intersection of `r=3+sintheta, r=2csctheta`?

Answer 1

Polar coordinates of points of intersection are `(3.56,34.16^@)` and `(3.56,145.84^@)`

Explanation:

Point of intersection of `r=3+sintheta` and `r=2csctheta`

will be given by `3+sintheta=2csctheta`

i.e. `3+sintheta=2/sintheta`

or `sin^2theta+3sintheta-2=0`

or `sintheta=(-3+-sqrt(3^2-4xx1xx(-2)))/2=(-3+-sqrt17)/2`

As `|sintheta|<=1`, `(-3-sqrt17)/2` is not permissible and hence

`sintheta=(-3+sqrt17)/2=0.5616` or `theta=34.16^@` or `180^@-34.16^@=145.84^@`

and `r=3+sintheta=3+(-3+sqrt17)/2=(3+sqrt17)/2~=3.56`

Hence, polar coordinates of points of intersection are `(3.56,34.16^@)` and `(3.56,145.84^@)`

These are depicted in rectangular coordinates as shown below.
graph{(y-2)(x^2+y^2-y-3sqrt(x^2+y^2))=0 [-10, 10, -5, 5]}