Question

Write down two different tests for symmetry with respect to

the polar axis. Find examples in which one test works and

the other fails. Which test do you prefer to use? Justify your

answer.

Short answer

The first test works for $r=\cos \theta$while fails for $r=\sin \theta$but the second test works for $r=\sin \theta$and fails for$r=\cos \theta$.

Step-by-step solution

Step 1. Given

A polar axis.

Step 2. Explanation

$$\begin{gathered} Symmetrytestwithrespecttopolaraxisi.e. \\ replace\theta with-\theta andsecondalyreplace\theta with\mathrm{\pi }-\mathrm{\theta } \\ \mathrm{therefore} \\ \mathrm{r}=\cos (-\mathrm{\theta }) \\ \mathrm{r}=\mathrm{cos\theta } \\ \mathrm{thus}\mathrm{first}\mathrm{test}\mathrm{works}\mathrm{for}\mathrm{r}=\mathrm{cos\theta } \\ \mathrm{Now}\mathrm{replace}\mathrm{\theta }\mathrm{with}\mathrm{\pi }-\mathrm{\theta }\mathrm{in}\mathrm{r}=\mathrm{cos\theta } \\ \mathrm{thus}, \\ \mathrm{r}=\cos (\mathrm{\pi }-\mathrm{\theta }) \\ =-\mathrm{cos\theta } \\ \mathrm{this}\mathrm{test}\mathrm{fails}\mathrm{for}\mathrm{r}=\mathrm{cos\theta } \end{gathered}$$

Step 3. Explanation

$$\begin{gathered} Nowconsiderr=\sin \theta \\ replace\theta with-\theta \\ r=\sin (-\theta ) \\ r=-\sin \theta \\ thustestispassed. \\ Nowreplace\theta with\mathrm{\pi }-\mathrm{\theta } \\ \mathrm{r}=\sin (\mathrm{\pi }-\mathrm{\theta }) \\ =\mathrm{sin\theta } \\ \mathrm{test}\mathrm{fails}\mathrm{for}\mathrm{r}=\mathrm{sin\theta } \end{gathered}$$