Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=\frac{1}{\sin \theta -\cos \theta }$

Short answer

The rectangular equation is$y-x=1$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=\frac{1}{\sin \theta -\cos \theta }$

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,$r=\frac{1}{\sin \theta -\cos \theta }$

Multiply by$(\sin \theta -\cos \theta )$ on both sides,

$\Rightarrow r(\sin \theta -\cos \theta )=1$

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,

$\begin{array}{l}\Rightarrow r\sin \theta -r\cos \theta =1\\ \Rightarrow y-x=1\end{array}$