# How do you convert the following to polar coordinates 2xy=1?

Jan 8, 2016

I think the question is convert to a polar equation. You use the rectangular conversion equations.

#### Explanation:

Based on the circular definitions of the trigonometric functions of sine and cosine you have the following:
$\sin \theta = \frac{y}{r}$ and $\cos \theta = \frac{x}{r}$.

therefore:

$y = r \sin \theta$ using algebra and
$x = r \cos \theta$ using algebra.
These are the conversion from rectangular $\left(x , y\right)$ to polar $\left(r , \theta\right)$

So in the equation $2 x y = 1$ you simply substitute.

$2 \cdot r \cos \theta \cdot r \sin \theta = 1$
$2 \cos \theta \cdot \sin \theta \cdot {r}^{2} = 1$
$\sin \left(2 \theta\right) \cdot {r}^{2} = 1$

${r}^{2} = {\csc}^{2} \left(2 \theta\right)$