Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=3(1-\sin \theta )$

Short answer

The rectangular equation is${(x^2+y^2+3y)}^2=9(x^2+y^2)$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=3(1-\sin \theta )$

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,$r=3(1-\sin \theta )$

Multiply by r on both sides,

$\Rightarrow r^2=3r-3r\sin \theta$

Formulae used,

$\begin{array}{l}{r}^2=x^2+y^2\\ x=r\cos \theta \\ y=r\sin \theta \end{array}$

Substitute the values in the formulae,

$\begin{array}{l}\Rightarrow x^2+y^2=3r-3y\\ \Rightarrow x^2+y^2+3y=3r\end{array}$

Squaring on both side,

$\begin{array}{l}\Rightarrow {(x^2+y^2+3y)}^2=9r^2\\ \because r^2=x^2+y^2\\ \Rightarrow {(x^2+y^2+3y)}^2=9(x^2+y^2)\end{array}$