# What does the polar equation cos2theta=1 represent?

Mar 30, 2017

$y = 0$

#### Explanation:

Observe that $\cos 2 \theta = 1$, when $\theta = 0$ or $\pi$. This is only on $x$-axis, whose equation is $y = 0$. Let us work it out the other way.

The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ is given by

$x = r \cos \theta$, $y = r \sin \theta$ and ${r}^{2} = {x}^{2} + {y}^{2}$

Hence $\cos 2 \theta = 1$ is ${\cos}^{2} \theta - {\sin}^{2} \theta = 1$

or ${x}^{2} / {r}^{2} - {y}^{2} / {r}^{2} = 1$

or $\left({x}^{2} - {y}^{2}\right) = {r}^{2} = {x}^{2} + {y}^{2}$

or $2 {y}^{2} = 0$ or $y = 0$ i.e. $x$-axis.