Chapter 6.5, Problem 39E

Solution

To analyze: The graph given by the polar equation $r=1-\cos \theta$ .

The equation represents a cardioid. Domain of the graph is $\left(-\infty ,\infty \right)$ . The range is $\left[0,2\right]$ . The graph is continuous and symmetric about the $x-$ axis. Maximum value of $r$ is $2$ . There are no asymptotes to the graph.

Given information:

The given polar equation is $r=1-\cos \theta$ .

Calculation:

Plot the graph for the polar equation.

  

Figure (1)

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

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

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

The range of $r=1-\cos \theta$ will be $\left[0,2\right]$ considering $1$ as maximum value and $-1$ as minimum value for $\cos \theta$ .

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

Since $r$ values are bounded between $\left[0,2\right]$ , so it is a bounded graph.

The maximum value of $r$ is $2$ .

There are no asymptotes to the graph.

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