Question

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

$r=\theta$

Short answer

The required equation is$y=\tan \sqrt{x^2+y^2}$.

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$r=\theta$

Step 2. Find the equation in rectangular coordinates.

$r=\theta$

Substitute $r=\sqrt{x^2+y^2}\mathrm{and}\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

$$\begin{gathered} \sqrt{x^2+y^2}={\tan }^{-1}\left(\frac{y}{x}\right) \\ \tan \sqrt{x^2+y^2}=\frac{y}{x} \\ x\tan \sqrt{x^2+y^2}=y \\ y=\tan \sqrt{x^2+y^2} \end{gathered}$$

Therefore, the equation in rectangular coordinates is$y=\tan \sqrt{x^2+y^2}$.