Question

Testing for Symmetry Test the polar equation for symmetry with respect to the polar axis, the pole, and the line$\theta =\pi /2$

.$r=2-\sin \theta$

Short answer

The polar equation$r=2-\sin \theta$. is not symmetric with respect to the polar axis, the pole and the line $\theta =\frac{\pi }{2}$.

Step-by-step solution

Step 1.Given information.

The given polar equation$r=2-\sin \theta$ .

Step 2.Finding given equation Symmetric or not.

• If in the polar equation $\text{r}=2-\sin \theta ,(r,\theta )$can be replaced by $(-r,\theta )$or$(r,\pi +\theta )$ ,then the graph is symmetric with respect to the pole.
• If in the polar equation $\text{r}=2-\sin \theta ,(r,\theta )$can be replaced by $(r,\pi -\theta )$or $(-r,-\theta )$ ,then the graph is symmetric with respect to the line.$\theta =\frac{\pi }{2}$

.

The given polar equation is$r=2-\sin \theta$.

Testing for the symmetry about the line$\theta =\frac{\pi }{2}$.

Replacing$(r,\theta )=(-r,-\theta )$ and simplifying the equation.
If the resulting equation is equal to the original equation then it is symmetry about the line.

$\begin{array}{l}\theta =\frac{\pi }{2}\\ -r=2-\sin (-\theta )\\ -r=2-(-\sin \theta )\\ -r=2+\sin \theta \\ \text{r}=-(2+\sin \theta )\\ \text{r}=-2-\sin \theta \end{array}$

The polar equation $r=2-\sin \theta$is not symmetric with respect to the line

$\theta =\frac{\pi }{2}$.

Step 3. Testing for symmetry aboupage polar axis.

Replacing$(r,\theta )=(r,-\theta )$ and simplifying the equation

$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

$\begin{array}{l}\Rightarrow \theta ={\tan }^{-1}\left(\frac{4}{3}\right)\\ \theta =0.295\pi ,1.295\pi \end{array}$

Since$\text{r>0}$, we consider$r=5$only and this occurs at$\theta =0.295\pi$.

and simplifying the equation

If the resulting equation is equal to the original equation$r=2-\sin \theta$then it is symmetry about the polar axis

$\begin{array}{l}\text{r}=2-\sin (-\theta )\\ r=2-(-\sin \theta )\\ \text{r}=2+\sin \theta \end{array}$

The polar equation $r=2-\sin \theta$is not symmetric with respect to the polar axis.

Step 4. Testing for symmetry about the pole.

Replacing$(r,\theta )=(-r,\theta )$and simplifying the equation

If the resulting equation is equal to the original equation$r=2-\sin \theta$then it is symmetry about the pole.

$\begin{array}{l}-r=2-\sin \theta \\ \text{r}=-(2-\sin \theta )\\ \text{r}=-2+\sin \theta \end{array}$

The polar equation $\text{r}=2+\sin \theta$ is not symmetric with respect to pole.