How do you convert #r^ 2 = 2 sin 2θ # to rectangular form?

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Answer 1 · 2016-09-21T04:39:49

#(x^2+y^2)^2=4xy#

Use the conversions
#r^2=x^2+y^2#
#x=rcostheta# or #costheta=x/r#
#y=rsintheta# or #sintheta=y/r#

#r^2=2sin(2theta)#

#r^2=2(2sinthetacostheta)color(white)(aaa)#Use the identity #sin2theta=2sinthetacostheta#
#r^2=4sinthetacostheta#

#x^2+y^2=4sinthetacosthetacolor(white)(aaaa)#Substitute #x^2+y^2# for #r^2#

#x^2+y^2=4*y/r*x/rcolor(white)(aaaaa)#Substitute #y/r# for #sintheta# and #x/r# for #costheta#

#x^2+y^2=(4xy)/r^2#

#x^2+y^2=(4xy)/(x^2+y^2)color(white)(aaaaaa)#Substitute #x^2+y^2# for #r^2#

#(x^2+y^2)^2=4xycolor(white)(aaaaaaa)#Multiply both sides by #(x^2+y^2)#