Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\sin 2\theta$

Short answer

The required equation is${\left(x^2+y^2\right)}^{\frac{3}{2}}=2xy$.

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$r=\sin 2\theta$

Step 2. Find the equation in rectangular coordinates.

$$\begin{gathered} r=\sin 2\theta \\ r=2\sin \theta \cos \theta \left[\sin 2\theta =2\sin \theta \cos \theta \right] \end{gathered}$$

First, multiply both sides byr,

$$\begin{gathered} r^2=2·r\sin \theta ·\cos \theta \\ {r}^2=2·y·\cos \theta \left[r\sin \theta =y\right] \end{gathered}$$

Again multiply both sides byr,

$$\begin{gathered} r·r^2=2·y·r\cos \theta \\ {r}^3=2·y·x\left[r\cos \theta =x\right] \\ {r}^3=2xy \\ {\left(\sqrt{x^2+y^2}\right)}^3=2xy\left[r^2=x^2+y^2\mathrm{and}r=\sqrt{x^2+y^2}\right] \\ {\left(x^2+y^2\right)}^{\frac{3}{2}}=2xy \end{gathered}$$

Therefore, the equation in rectangular coordinates is${\left(x^2+y^2\right)}^{\frac{3}{2}}=2xy$.