Write the equation #r=2cos2theta# in polar form in Cartesian or rectangular form?

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Answer 1 · 2017-12-06T16:10:47

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

The relation between polar coordinates #(r,theta)# and rectangular or Cartesian coordinates #(x,y)# is given by

#x=rcostheta# and #y=rsintheta# i.e. #x^2+y^2=r^2#

Hence #r=2cos2theta=2(cos^2theta-sin^2theta)#

or #r^3=2(r^2cos^2theta-r^2sin^2theta)#

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

The graph appears as shown below.

graph{(x^2+y^2)^(3/2)=2(x^2-y^2) [-2.5, 2.5, -1.25, 1.25]}