Q. 13 - Calculus

Question

Which kind(s) of symmetry does the rose$r = \cos 5\theta$ have? How many petals does this curve have?

 Which kind(s) of symmetry does the rose $r = \sin 8\theta$ have? How many petals does this curve have? 

Step-by-Step Solution

Answer

The rose $r=\cos 5\theta$ is symmetrical about $x-$axis and it has five petals.

The rose $r=\sin 8\theta$ is not symmetrical about the x-axis and y-axis. It has $16$petals..

Step 1. Given information

The equation of roses:

$$\begin{gathered} r=\cos 5\theta \\ r=\sin 8\theta \end{gathered}$$

Step 2. Find the symmetry and number of petals of r = cos 5 θ .

$$\begin{gathered} r\left(\theta \right)=\cos 5\theta \\ r(-\theta )=\cos (-5\theta ) \\ r(-\theta )=\cos 5\theta \\ r(-\theta )=r\left(\theta \right) \end{gathered}$$

So this is symmetrical about the x-axis.

$$\begin{gathered} r(\mathrm{\pi }-\mathrm{\theta })=\cos 5(\mathrm{\pi }-\mathrm{\theta }) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=\cos 5(\mathrm{\pi }-\mathrm{\theta }) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=\cos (5\mathrm{\pi }-5\mathrm{\theta }) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=\cos (4\mathrm{\pi }+\mathrm{\pi }-5\mathrm{\theta }) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=\cos (\mathrm{\pi }-5\mathrm{\theta }) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=-\cos \left(5\mathrm{\theta }\right) \\ \mathrm{r}(\mathrm{\pi }-\mathrm{\theta })=-\mathrm{r}\left(\mathrm{\theta }\right) \end{gathered}$$

This is not symmetrical about y-axis.

The graph of the function is 

It has 5 petals.

Step 3. Find the symmetry and number of petals of r = sin 8 θ .

$$\begin{gathered} r\left(\theta \right)=\sin 8\theta \\ r(-\theta )=\sin (-8\theta ) \\ r(-\theta )=-\sin 8\theta \\ r(-\theta )=-r\left(\theta \right) \end{gathered}$$

So the graph is not symmetrical about the x-axis.

$$\begin{gathered} r(\pi -\theta )=\sin 8(\pi -\theta ) \\ r(\pi -\theta )=\sin 8(\pi -\theta ) \\ r(\pi -\theta )=-\sin \left(8\theta \right) \\ r(\pi -\theta )=-r\left(\theta \right) \end{gathered}$$

So the graph is not symmetrical about the y-axis.

So the graph is not symmetrical about the origin.

It has 16 petals.