How do you convert #r=1+2sin θ# into a rectangular equations?

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Answer 1 · 2016-02-09T05:04:54

#x^2 + y^2 = (x^2 + y^2 - 2y)^2#

First, multiply both sides by #r#.

#r^2 = r + 2 r sin theta#

Use the following identities

#r^2 = x^2 + y^2#
#y = r sin theta#

Substitute the rectangle forms in

#x^2 + y^2 = r + 2y#

Subtract #2y# on both sides to make #r# the subject

#r = x^2 + y^2 - 2y#

Square both sides to get the #r^2# term

#r^2 = (x^2 + y^2 - 2y)^2#

Use the above identity again to remove the #r^2#

#x^2 + y^2 = (x^2 + y^2 - 2y)^2#