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.
