Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$\cos 2\theta =1$

Short answer

The rectangular equation is$y=0$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$\cos 2\theta =1$

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,$\cos 2\theta =1$

$\begin{array}{l}2{\cos }^2\theta -1=1\\ \Rightarrow 2{\cos }^2\theta =2\\ \Rightarrow {\cos }^2\theta =1\\ \Rightarrow r^2{\cos }^2\theta =r^2\\ \text{Invalid <m:msup> element}\Rightarrow =x^2+y^2\\ \Rightarrow x^2=x^2+y^2\\ \Rightarrow y^2=0\\ \Rightarrow y=0\end{array}$