How do you convert #r=1-2cos2t# to rectangular form?

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Answer 1 · 2016-07-25T03:30:30

#(x^2+y^2)^(3/2)=3y^2-x^2#

The relation between polar coordinates #(r,t)# and rectangular coordinates #(x,y)# are given by #x=rcost# and #y=rsint# and hence #r=sqrt(x^2+y^2)#

Thus #r=1-2cos2t# can be written as

#r=1-2(2cos^2t-1)=3-4cos^2t# or

#r=3-4cos^2t=3-4x^2/r^2# or

#r^3=3r^2-4x^2# or

#(x^2+y^2)^(3/2)=3x^2+3y^2-4x^2#

#(x^2+y^2)^(3/2)=3y^2-x^2#

graph{(x^2+y^2)^(3/2)=3y^2-x^2 [-10, 10, -5, 5]}