The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). xy = 1 Answer choices: A) r sin 2θ = 2 B) 2r sin θ cos θ = 1 C) r^2 sin 2θ = 2 D) 2r^2 sin θ cos θ = 1 Could you explain to me how to solve this?

To express the equation $xy = 1$ in polar coordinates $r$ and $\theta$, you can use the following steps:

1. Convert rectangular coordinates to polar coordinates using the relations:

   $$\begin{align*} x &= r \cos \theta \\ y &= r \sin \theta \end{align*}$$

2. Substitute $x = r \cos \theta$ and $y = r \sin \theta$ into the given equation $xy = 1$.

3. Solve for $r$ and $\theta$ in terms of $r$ and $\theta$.

Applying these steps:

$xy = 1$ $(r \cos \theta)(r \sin \theta) = 1$ $r^2 \cos \theta \sin \theta = 1$

Using the trigonometric identity $2 \sin \theta \cos \theta = \sin 2\theta$, the equation can be rewritten as:

$r^2 \sin 2\theta = 2$

Therefore, the equation in polar coordinates is $r^2 \sin 2\theta = 2$, which corresponds to option C).