How do you convert #r=2sin theta + cos theta# into rectangular form?

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Answer 1 · 2017-02-25T16:29:14

The equation is #(x-1/2)^2+(y-1)^2=5/4#

To convert from polar coordinates #(r, theta)# to rectangular coordinates #(x,y)#, we use the following equations

#x=rcostheta#, #=>#, #costheta=x/r#

#y=rsintheta#, #=>#, #sintheta=y/r#

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

Therefore,

#r=2sintheta+costheta#

#r=2*y/r+x/r#

#r^2=2y+x#

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

#x^2-x+y^2-2y=0#

#x^2-x+1/4+y^2-2y+1=1+1/4#

#(x-1/2)^2+(y-1)^2=5/4#

This is the equation of a circle, center #(1/2,1)# and radius #=sqrt5/2#