Question

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

$r=\tan \theta$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$r=\tan \theta$

Step 2. Find the equation in rectangular coordinates.

$$\begin{gathered} r=\tan \theta \\ r=\frac{\sin \theta }{\cos \theta } \end{gathered}$$

multiply both sides byr,

$$\begin{gathered} r^2=\frac{r\sin \theta }{r\cos \theta } \\ {r}^2=\frac{y}{x}\left[\mathrm{Since}r\sin \theta =y\mathrm{and}r\cos \theta =x\right] \\ {x}^2+y^2=\frac{y}{x}\left[r^2=x^2+y^2\right] \end{gathered}$$

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