Question

Finish Example 1 by showing that $r=\cos 2\theta$is symmetrical with respect to the horizontal axis.

Short answer

It has been proved that the curve is symmetrical about the horizontal axis.

Step-by-step solution

Step 1. Given information

Equation of polar curve:

$r\left(\theta \right)=\cos \left(2\theta \right)$

Step 2. Show the symmetry with respect to the horizontal axis.

If $r\left(-\theta \right)=r\left(\theta \right)$then the curve is symmetrical about the x-axis.

$$\begin{gathered} r\left(\theta \right)=\cos \left(2\theta \right) \\ r\left(-\theta \right)=\cos 2\left(-\theta \right) \\ r\left(-\theta \right)=\cos \left(-2\theta \right) \\ r\left(-\theta \right)=\cos \left(2\theta \right) \\ r\left(-\theta \right)=r\left(\theta \right) \end{gathered}$$

Hence the given curve is symmetrical about the horizontal axis.