# Trigonometryconvert r = 5sin(2θ) to rectangular coordinates

#### karush

##### Well-known member
convert $$\displaystyle r=5\sin{2\theta}$$ 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?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Re: convert r=5sin2\theta to rectangular coordinates

$\dfrac{a^2}{\sqrt{a}}=a^{3/2}$ for every $a>0$.

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#### Klaas van Aarsen

##### MHB Seeker
Staff member
Re: convert r=5sin2\theta to rectangular coordinates

convert $$\displaystyle r=5\sin{2\theta}$$ 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:
$$(x^2+y^2) \sqrt{x^2+y^2} = 10 \cdot y \cdot x$$
$$(x^2+y^2)^{3/2} = 10 xy$$

#### karush

##### Well-known member
Re: convert r=5sin2\theta to rectangular coordinates

I should of seen that...

at least my basic steps were ok..