Q 59. - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given information

$r = \cos n\theta$

Step 2: Calculation

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

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

That is show that for every $(r,\theta )$ the point $(r,\pi -\theta )$ is on the graph.

Let $(r,\theta )$ be any point on the graph and $f\left(\theta \right)=\cos n\theta$ where $n$ is an even integer.

By the definition of symmetry,

If every point $(r,\theta )$ on the graph also has the point $(r,-\theta )$ or $(r,\pi -\theta )$ the curve is symmetric with regard to the $y-$axis.

Step 3: Calculation

When $n$ is a positive odd integer for the polar rose $r=\cos n\theta$ traced twice on the interval $[0,2\pi ]$

Take the equation $f\left(\theta \right)=\cos n\theta$ where $n$ is even

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

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

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

Hence it is proved.