Cartesian Equation for Polar Curve

Question

r = 4tan(theta)sec(theta)

How do you convert this into y=?

Answer 1

r =4tan(theta)sec(theta) =4y/x)(r/x)

1= 4y/x^2

x^2 = 4y

the equation is for an upward opening parabola with vertex at the origin

given standard conversion equations

x=rcos(theta)

and y = rsin(theta)

sometimes it's easier to convert trig functions to sines and cosines, especially if you don't know what else to do.

tan(theta) = sin(theta)/cos(theta),

sec(theta) = 1/cos(theta)

r =4sin(theta)/cos(theta)(1/cos(theta)

r= 4y/x(r/x)

1=4y/x^2

4y = x^2

Answer 2

Convert tangent and secant to sine and cosine and simplify.