Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r^2=\tan \theta$

Short answer

The rectangular equation is$x^3+x{y}^2=y$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r^2=\tan \theta$

Step 2. Concept used

To convert rectangular coordinates to polar coordinates, use the formulae below-

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

Convert polar coordinates using the same formulas-

$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Substitute the value in the formulae. And simplify them.

Step 3. Calculation

The polar equation is given as,$r^2=\tan \theta$

By using above formulae,

$\begin{array}{l}{x}^2+y^2=\frac{y}{x}\\ x^3+x{y}^2=y\end{array}$