# Thread: Converting Polar equations into rectangular equations:

1. ## Converting Polar equations into rectangular equations:

How do I convert r = 5sin(x) into an equivalent rectangular equation?

2. Originally Posted by Neversh
How do I convert r = 5sin(x) into an equivalent rectangular equation?
you mean

$\displaystyle r=5sin(\theta)$ to rectangular

you should know that

$\displaystyle x=r cos(\theta)$

$\displaystyle y=r sin(\theta)$

$\displaystyle \sqrt{x^2+ y^2 } = r.....x^2+y^2=r^2$

use them to find the equation try

3. Hello, Neversh!

You know the conversion, right?

. . $\displaystyle \begin{array}{c}r\cos\theta \:=\:x \\ r\sin\theta \:=\:y \\ r^2 \:=\:x^2+y^2 \end{array}$

How do I convert $\displaystyle r = 5\sin\theta$ into an equivalent rectangular equation?

We have: .$\displaystyle r \:=\:5\sin\theta$

$\displaystyle \text{Multiply by }r\!:\;\;\underbrace{r^2} \:=\:5\underbrace{r\sin\theta}$
. - . - . . . . . . .$\displaystyle \uparrow\qquad\qquad\: \uparrow$
. . . . . . . . . .$\displaystyle ^{x^2+y^2}\qquad\quad\;\; ^y$

And we have: .$\displaystyle x^2+y^2 \:=\:5y$

4. Originally Posted by Neversh
How do I convert r = 5sin(x) into an equivalent rectangular equation?
First multiply both sides by r to get

$\displaystyle r^2=5r\sin(x)$

Now we know that $\displaystyle r^2=x^2+y^2$ and

$\displaystyle r\sin(x)=y$

Now we get

$\displaystyle x^2+y^2=5y \iff x^2+y^2-5y=0 \iff x^2+y^2-5y+\frac{25}{4}=\frac{25}{4}$

So finally we get

$\displaystyle x^2+\left(y-\frac{5}{2} \right)^2=\left( \frac{5}{2}\right)^2$

Edit too slow haha