MHB Convert r = 5sin(2θ) to rectangular coordinates

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The discussion focuses on converting the polar equation r = 5sin(2θ) to rectangular coordinates. The derived equation is (x² + y²)^(3/2) = 10xy. The conversion process involves multiplying both sides by r, substituting r² with x² + y², and using the identity for sin(2θ). Participants clarify the derivation of (x² + y²)^(3/2) and confirm the correctness of the steps taken. The final equation effectively represents the original polar equation in rectangular form.
karush
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convert $$r=5\sin{2\theta}$$ to rectangular coordinates

the ans to this is $\left(x^2+y^2\right)^{3/2}=10xy$

however... multiply both sides by $r$ to get $r^2=5\cdot r \cdot \sin{2\theta}$

then substitute $r^2$ with $x^2+y^2$
and $\sin{2\theta}$ with $2\sin\theta\cos\theta$
and divide each side by $r$

$$\frac{x^2+y^2}{\sqrt{x^2+y^2}}=10xy$$

how is $\left(x^2+y^2\right)^{3/2}$ derived?
 
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Re: convert r=5sin2\theta to rectangular coordinates

$\dfrac{a^2}{\sqrt{a}}=a^{3/2}$ for every $a>0$.
 
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Re: convert r=5sin2\theta to rectangular coordinates

karush said:
convert $$r=5\sin{2\theta}$$ to rectangular coordinates

the ans to this is $\left(x^2+y^2\right)^{3/2}=10xy$

however... multiply both sides by $r$ to get $r^2=5\cdot r \cdot \sin{2\theta}$

then substitute $r^2$ with $x^2+y^2$
and $\sin{2\theta}$ with $2\sin\theta\cos\theta$

That is:
$$x^2+y^2 = 5\cdot r \cdot 2\sin\theta\cos\theta$$
$$x^2+y^2 = 10 \cdot \sin\theta \cdot r\cos\theta$$
and divide each side by $r$

$$\frac{x^2+y^2}{\sqrt{x^2+y^2}}=10xy$$

how is $\left(x^2+y^2\right)^{3/2}$ derived?

Let's multiply by $r$ instead of divide by it.
$$(x^2+y^2) r = 10 \cdot r\sin\theta \cdot r\cos\theta$$
Now make the substitutions:
$$(x^2+y^2) \sqrt{x^2+y^2} = 10 \cdot y \cdot x$$
$$(x^2+y^2)^{3/2} = 10 xy$$
 
Re: convert r=5sin2\theta to rectangular coordinates

I :)should of seen that...

at least my basic steps were ok..
 
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