Question

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

$r^2=\sin \theta$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$r^2=\sin \theta$

Step 2. Find the equation in rectangular coordinates.

$r^2=\sin \theta$

First, multiply both sides byr,

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

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