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.

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