91Ó°ÊÓ

$r=\cos n\mathrm{θ}$

Consider the polar equation $r=\cos n\mathrm{θ}$

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

That is show that for every $(r,\mathrm{θ})$the point $(r,\mathrm{π}-\mathrm{θ})$is on the graph.

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

By the definition of symmetry,

If every point $(r,\mathrm{θ})$ on the graph also has the point $(r,-\mathrm{θ})$ or $(r,\mathrm{π}-\mathrm{θ})$ the curve is symmetric with regard to the $y-$axis.

When $n$is a positive odd integer for the polar rose $r=\cos n\mathrm{θ}$traced twice on the interval $[0,2\mathrm{π}]$

Take the equation $f\left(\mathrm{θ}\right)=\cos n\mathrm{θ}$where $n$is even

$$\begin{gathered} f(\mathrm{π}-\mathrm{θ})=\cos n(\mathrm{π}-\mathrm{θ}) \\ =\cos n\mathrm{θ} \end{gathered}$$

As a result, the point $(r,\mathrm{θ})$is also on the graph, as is the point $(r,\mathrm{π}-\mathrm{θ})$

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

Hence it is proved.