Question

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

Short answer

The function $r=\sin n\theta$ is symmetric about the $y-$axis for all values of $n$

Step-by-step solution

Given information

$r=\sin n\theta$

Calculation

Consider the polar coordinate system function $r=\sin n\theta$

Here, $n$is any integer.

The objective is to prove that the graph of the function is symmetric about the $y-$axis for every integer $n$

The graph in a polar coordinate system is symmetric about the $y-$axis if it is invariant with respect to the transformation $\theta \to \pi -\theta$ or $\theta \to -\theta$

Calculation

Consider the transformation $\theta \to \pi -\theta$ in the function $r=\sin n\theta$

$$\begin{gathered} r=\sin n(\pi -\theta ) \\ =\sin (n\pi -n\theta ) \end{gathered}$$

For any integer $n$ the angle $(n\pi -n\theta )$ lies in the second quadrant and in the second quadrant $\sin$ function is symmetric.

That is, $r=\sin n\theta$

Thus, the function does not change under the transformation $\theta \to \pi -\theta$

Hence, the function $r=\sin n\theta$ is symmetric about the $y-$axis for all values of $n$