Chapter 6.5, Problem 40E

Solution

To analyze: The graph given by the polar equation $r=2+\sin \theta$ .

The equation represents a convex limaçon. Domain of the graph is $\left(-\infty ,\infty \right)$ . The range is $\left[1,3\right]$ . The graph is continuous and symmetric about the $y-$ axis. Maximum value of $r$ is $3$ . There are no asymptotes to the graph.

Given information:

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

Calculation:

Plot the graph for the polar equation.

  

Figure (1)

From the figure (1), it can be observed that the equation represents a convex limaçon.

The equation is defined for all real numbers of $\theta$ in the domain $\left(-\infty ,\infty \right)$ .

Range of $\sin \theta$ is, $-1\le \sin \theta \le 1$ .

The range of $r=2+\sin \theta$ will be $\left[1,3\right]$ considering $1$ as maximum value and $-1$ as minimum value for $\sin \theta$ .

The graph is continuous. The graph is symmetric about the $y-$ axis.

Since $r$ values are bounded between $\left[1,3\right]$ , so it is a bounded graph.

The maximum value of $r$ is $3$ .

There are no asymptotes to the graph.

Therefore, the polar equation $r=2+\sin \theta$ is analyzed with the help of figure (1).