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

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Answer 1 · 2017-07-21T08:35:01

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

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]}