Chapter 6.5, Problem 49E

Solution

To find: The two equations from the given three equations $r_1=1+3\sin \theta$ , $r_2=-1+3\sin \theta$ , and $r_3=1-3\sin \theta$ that have identical graphs. Also analyze, the two identical graphs for increasing values of $\theta$ .

The graph of $r_1=1+3\sin \theta$ and $r_2=-1+3\sin \theta$ is identical. $r_1$ is traced from $\left(1,0\right)$ in the clockwise manner and $r_2$ is traced from $\left(-1,0\right)$ in the counter-clockwise manner.

Given information:

The given equations are $r_1=1+3\sin \theta$ , $r_2=-1+3\sin \theta$ , and $r_3=1-3\sin \theta$ .

Calculation:

Graph for the polar equation $r_1=1+3\sin \theta$ .

  

Figure (1)

Graph for the polar equation $r_2=-1+3\sin \theta$ .

  

Figure (2)

Graph for the polar equation $r_3=1-3\sin \theta$ .

  

Figure (3)

From the graphs, the curve of $r_1=1+3\sin \theta$ and $r_2=-1+3\sin \theta$ is identical.

  $r_1=1+3\sin \theta$ is traced from $\left(1,0\right)$ in the clockwise manner.

  $r_2=-1+3\sin \theta$ is traced from $\left(-1,0\right)$ in the counter-clockwise manner.