How do you graph the polar equation $r^{2} = 9 sin 2 θ$ $r^2=9sin2theta$ ?

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How do you graph the polar equation $r^{2} = 9 sin 2 θ$ $r^2=9sin2theta$ ?

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Answer 1 · 2016-11-08T19:25:11

${(x^{2} + y^{2})}^{2} - 18 x y = 0$ $(x^2+y^2)^2-18xy=0$

Explanation:

$x = ρ cos θ$ $x=rhocostheta$ from which $cos θ = \frac{x}{ρ}$ $costheta=x/rho$
$y = ρ sin θ$ $y=rhosintheta$ from which $sin θ = \frac{y}{ρ}$ $sintheta=y/rho$
$ρ = \sqrt[2]{x^{2} + y^{2}}$ $rho=root2(x^2+y^2)$
$ρ^{2} = x^{2} + y^{2}$ $rho^2=x^2+y^2$
$ρ^{2} = 18 sin θ cos θ =$ $rho^2=18sinthetacostheta=$
$ρ^{2} = 18 \frac{x y}{ρ^{2}}$ $rho^2=18(xy)/rho^2$
$ρ^{4} = 18 x y$ $rho^4=18xy$
${(x^{2} + y^{2})}^{2} = 18 x y$ $(x^2+y^2)^2=18xy$

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How do you graph the polar equation $r^{2} = 9 sin 2 θ$ $r^2=9sin2theta$ ?

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