How do you convert #y=x^2# into polar form?

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Answer 1 · 2018-03-11T10:29:15

#color(Blue)(r=tan(theta)*sec(theta)# in the Polar form.

Given:

Cartesian form #color(blue)(y = f(x) = x^2#

We must convert this into equivalent polar form.

To convert to polar form use

#color(brown)(x = r cos theta# and

#color(brown)(y = r sin theta#

Consider the given Cartesian form

#y = x^2#

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

#r sin theta = r^2 cos^2theta#

Divide both sides by #r#

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

#(cancel(r) sin theta)/cancel(r) = (cancel(r^2)^color(red)r cos^2theta)/cancel(r#

#sin theta = r cos^2 theta#

Divide both sides by #cos^2 theta#

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

#(sin theta)/cos^2theta = (r cancel(cos^2 theta))/cancel(cos^2theta)#

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

#r = (sin theta)/(cos theta*cos theta)#

#r = (sin theta)/(cos theta)*1/cos theta#

#color(Blue)(r=tan(theta)*sec(theta)#

Required answer in the Polar form.