Rewrite the rectangular equation to a polar equation, y=2x^2, what would r(theta) equal?

Question

y=2x^2

r(theta)= ?

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Answer 1 · 2018-04-08T00:27:35

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

$y=2x^2$

$2y^2+y=2x^2+2y^2$

$2(r(theta)sin(theta))^2+r(theta)sin(theta)=2(r(theta))^2$

$2(r(theta))^2sin^2(theta)+r(theta)sin(theta)=2(r(theta))^2$

$2r(theta)sin^2(theta)+sin(theta)=2r(theta)$

$2r(theta)sin^2(theta)-2r(theta)=-sin(theta)$

$r(theta)(2sin^2(theta)-2)=-sin(theta)$

$r(theta)=-sin(theta)/(2sin^2(theta)-2)$

$r(theta)=sin(theta)/(2-2sin^2(theta))$

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

(note: i think you can divide both sides by $r(theta)$ in step 5 because the function $r(theta)$ passes through the origin $(r=0)$ anyway)