Question

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$\mathrm{r}=4\cos 3\mathrm{\theta }$.

Answer 1

Determining the symmetrical of the graph:

Test for symmetrical :

If $\mathrm{f}(\mathrm{r},\mathrm{\theta })=\mathrm{f}(-\mathrm{r},-\mathrm{\theta })$, symmetrical to the pole or the origin

If $\mathrm{f}(\mathrm{r},\mathrm{\theta })=\mathrm{f}(\mathrm{r},-\mathrm{\theta })$, symmetrical to the polar axis or the $x$ axis

If $\mathrm{f}(\mathrm{r},\mathrm{\theta })=\mathrm{f}(-\mathrm{r},\mathrm{\theta })$, symmetrical to the $y$axis.

Step-1: About the $x$- axis:

$$\begin{gathered} \mathrm{f}(\mathrm{r},\mathrm{\theta })\Rightarrow \mathrm{r}=4\mathrm{cos\theta } \\ \mathrm{f}(\mathrm{r},-\mathrm{\theta })\Rightarrow \mathrm{r}=4\cos (-\mathrm{\theta }) \\ =4\cos \mathrm{\theta }\left[\cos \right(-\mathrm{\theta })=\mathrm{cos\theta }] \\ \therefore \mathrm{f}(\mathrm{r},\mathrm{\theta })=\mathrm{f}(\mathrm{r},-\mathrm{\theta }) \end{gathered}$$

So, the graph is symmetrical about the $x$-axis:

Step-2: About origin:

$$\begin{gathered} \mathrm{f}(\mathrm{r},\mathrm{\theta })\Rightarrow \mathrm{r}=4\mathrm{cos\theta } \\ \mathrm{f}(-\mathrm{r},-\mathrm{\theta })\Rightarrow -\mathrm{r}=4\cos (-\mathrm{\theta }) \\ \Rightarrow r=-4\cos \mathrm{\theta }\left[\cos \right(-\mathrm{\theta })=\mathrm{cos\theta }] \\ \therefore \mathrm{f}(\mathrm{r},\mathrm{\theta })\ne \mathrm{f}(\mathrm{r},-\mathrm{\theta }) \end{gathered}$$

Hence the graph is not symmetrical about origin.

Step-3: About $y$ axis:

$$\begin{gathered} \mathrm{f}(\mathrm{r},\mathrm{\theta })\Rightarrow \mathrm{r}=4\mathrm{cos\theta } \\ \mathrm{f}(-\mathrm{r},\mathrm{\theta })\Rightarrow -\mathrm{r}=4\cos \left(\mathrm{\theta }\right) \\ \Rightarrow r=-4\cos \mathrm{\theta } \\ \therefore \mathrm{f}(\mathrm{r},\mathrm{\theta })\ne \mathrm{f}(-\mathrm{r},\mathrm{\theta }) \end{gathered}$$

Hence, the graph is not symmetrical $y$-axis.

Therefore, the graph is symmetrical about the $x$-axis only.