Q 41. - Calculus

Question

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

$r=\sin 2\theta$

Step-by-Step Solution

Answer

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

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 by r,

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

Again multiply both sides by r,

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

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