How do I convert r = 5sin(x) into an equivalent rectangular equation?
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: .$\displaystyle x^2+y^2 \:=\:5y$
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