Chapter 9.2, Problem 10AYU

To determine

To fill: What has to be replaced to test whether the graph of a polar equation may be symmetric with respect to line $\theta = \frac{\pi }{2}$ of $\theta$.

Answer to Problem 10AYU

Solution:

To test whether the graph of a polar equation may be symmetric with respect to to line $\theta = \frac{\pi }{2}$ instead of $\pi - \theta$.

Given:

Statement asked the replacement term to test whether the graph of a polar equation may be symmetric with respect to the line $\theta = \frac{\pi }{2}$ instead $\theta$.

Explanation of Solution

Test for symmetry:

Symmetry with respect to the line $\theta = \frac{\pi }{2}\left(y - Axis\right)$:

In a polar equation, replace $$ \theta $$ by $\pi - \theta$. If an equivalent equation results, the graph is symmetric with respect to the line $\theta = \frac{\pi }{2}$.