How do you convert # x^2 + y^2 = 4y# to polar form?

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Answer 1 · 2018-04-22T13:46:44

The polar equation is #r=4sintheta#

Apply the following to convert from rectangular coordinates #(x,y)# to polar coordinates #(r,theta)# :

#{(x=rcostheta),(y=rsintheta),(x^2+y^2=r^2), (theta=arctan(y/x)):}#

Here, we have

#x^2+y^2=4y#

#r^2=4rsintheta#

As #r!=0#

#r=4sintheta#

Answer 2 · 2018-04-22T13:48:57

#r=4sintheta#

The conversion from Rectangular to Polar:
#x=rcostheta#
#y=rsintheta#

Substitute for #x# and #y#:
#(rcostheta)^2+(rsintheta)^2= 4(rsintheta)#

#r^2cos^2theta+r^2sin^2theta= 4rsintheta#

Factor out the #r^2#:
#r^2(cos^2theta+sin^2theta)= 4rsintheta#

Apply pythagorean identity:
#sin^2theta+cos^2theta=1#
#r^2(1)= 4rsintheta#

Set the expression equal to 0:
#r^2-4rsintheta=0#

Factor out the #r#:
#r(r-4sintheta)=0#

At this point either #r=0# or #r-4sintheta=0#, let's solve the second to get a meaningful answer:

#r=4sintheta#