Question

Prove that, for every integer $n$ the graph of $r=\cos n\theta$ is symmetrical with respect to the $x-$axis.

Short answer

The polar equation $r=\cos n\theta$ is symmetric with respect to$x-$axis.

Step-by-step solution

Given information

$r=\cos n\theta$

Calculation

Consider the polar equation $r=\cos n\theta$

The goal is to show that the equation is symmetric around the$x-$axis.

If for any point $(r,\theta )$ on the graph, the point $(r,-\theta )$ is also on the graph, the curve is symmetric with respect to the $x-$axis.

By the definition of symmetry,

A curve is symmetric with respect to the$x-$axis if, for every point $(r,\theta )$ on the graph, the point $(r,-\theta )$ is also on the graph.

Calculation

Take the equation $f\left(\theta \right)=\cos n\theta$where $n$any integer

$$\begin{gathered} f(-\theta )=\cos n(-\theta ) \\ =\cos n\theta \end{gathered}$$

As a result, the point $(r,\theta )$is also on the graph, as is the point $(r,-\theta )$

As a result, with regard to the $x-$axis, the polar equation $r=\cos n\theta$ is symmetric.

Hence it is proved.