How do you convert $r = sin 2 (θ)$ $r=sin2(theta)$ in rectangular form?

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How do you convert $r = sin 2 (θ)$ $r=sin2(theta)$ in rectangular form?

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Answer 1 · 2018-05-13T14:45:13

${(x^{2} + y^{2})}^{3} = 4 x^{2} y^{2}$ $(x^2 + y^2 ) ^3 = 4x^2y^2$

Explanation:

$r = sin 2 θ$ $r = sin2theta$

$\Rightarrow r = 2 sin θ cos θ$ $=> r = 2sintheta costheta$

We know $r cos θ = x$ $rcos theta = x$

$\Rightarrow cos θ = \frac{x}{r} = \frac{x}{\sqrt{x^{2} + y^{2}}}$ $=> cos theta = x / r = x / ( sqrt(x^2 + y^2 )$

$\Rightarrow \sqrt{x^{2} + y^{2}} = 2 \cdot \frac{x}{\sqrt{x^{2} + y^{2}}} \cdot \frac{y}{\sqrt{x^{2} + y^{2}}}$ $=> sqrt(x^2 + y^2) = 2 * x/sqrt(x^2 +y^2) * y/sqrt(x^2 + y^2 )$

$\Rightarrow \sqrt{x^{2} + y^{2}} = \frac{2 x y}{x^{2} + y^{2}}$ $=> sqrt(x^2 + y^2 ) = (2xy)/(x^2 + y^2 )$

$\Rightarrow {(x^{2} + y^{2})}^{\frac{3}{2}} = 2 x y$ $=> (x^2 + y^2) ^ (3/2) = 2xy$

$\Rightarrow {(x^{2} + y^{2})}^{3} = 4 x^{2} y^{2}$ $=> (x^2 + y^2 ) ^3 = 4x^2y^2$

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How do you convert $r = sin 2 (θ)$ $r=sin2(theta)$ in rectangular form?

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