Just to remember the basics of polar coordinates:
Polar Coordinates
Known that #sec theta= 1/cos theta#, the expression in polar coordinates may be so rewritten:
#r=4*tan theta(1/cos theta)#
Toward the convertion we know that:
#r=sqrt(x^2+y^2)#
#theta=arc tan(y/x)#
Then #tan theta=tan (arc tan (y/x))=y/x#
To complete the convertion we only need to determine #cos theta# in function of x and y:
#tan theta = y/x# => #sin theta/cos theta =y/x# => #sqrt (1-cos^2 theta)/cos theta =y/x# => #(1-cos^2 theta)/cos^2 theta=y^2/x^2# => #x^2-x^2*cos^2 theta = y^2*cos^2 theta# => #cos^2 theta*(x^2+y^2)=x^2# => #cos theta =x/sqrt(x^2+y^2)#
Substituting #r#, #tan theta# and #cos theta#, by the corresponding functions in x and y, the original expression becomes:
#sqrt(x^2+y^2)=4.(y/x)(1/(x/sqrt(x^2+y^2)))# => #cancel(sqrt(x^2+y^2))*(x/cancel(sqrt(x^2+y^2)))=4.(y/x)# => #x^2=4y#
Testing the result (or reconverting it to polar coordinates):
#x^2=4y# => #(r*cos theta)^2=4*r*sin theta# => #r ^cancel(2)*cos^2 theta=4*cancel(r)*sin theta# => #r=4(sin theta/cos theta)(1/(cos theta))# [This expression is equivalent to the original expression. Thus the resulting expression (in x and y) is correct.]
