1. quick polar coordinates question

In 3r = sin A, r and A represent polar coordinates, write each polar equation as an equation in rectangular coordinates (x,y).

thanks

2. $\displaystyle r=\sqrt{x^2+y^2}$ and $\displaystyle \displaystyle \tan A=\frac{y}{x}$.
We have $\displaystyle \displaystyle \sin A=\frac{\tan A}{\sqrt{1+\tan ^2A}}=\frac{y}{\sqrt{x^2+y^2}}$.
Replacing $\displaystyle r$ and $\displaystyle \sin A$ in the given relation, we have
$\displaystyle \displaystyle 3\sqrt{x^2+y^2}=\frac{y}{\sqrt{x^2+y^2}}\Leftright arrow 3(x^2+y^2)=y$

3. Hello, Andy!

I assume you know the conversion formulas:
. . $\displaystyle x \,= \,r\cos\theta,\;\;y \,= \,r\sin\theta,\;\;x^2 + y^2 \:=\:r^2$

In $\displaystyle 3r \:= \:\sin\theta,\;\theta$ and $\displaystyle r$ represent polar coordinates.
Write the equation in rectangular coordinates $\displaystyle (x,y)$
$\displaystyle \text{We have: }3r \;=\;\sin\theta$

$\displaystyle \text{Multiply both sides by }r\!:\;\;3r^2 \;=\;\underbrace{r\sin\theta}$
. . . . . . . . . . . . . .$\displaystyle 3\overbrace{(x^2+y^2)} \:=\:y$

4. thanks