test for symmetry with respect to (a) the polar axis, (b) the line $\theta=\frac{pi}{2}$, and (c) the pole…

test for symmetry with respect to (a) the polar axis, (b) the line $\theta=\frac{pi}{2}$, and (c) the pole for the given polar equation. $r = 2cos\theta$ a. is the graph of the polar equation symmetric with respect to the polar axis? a. yes. b. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis. c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.

test for symmetry with respect to (a) the polar axis, (b) the line $\theta=\frac{pi}{2}$, and (c) the pole for the given polar equation. $r = 2cos\theta$ a. is the graph of the polar equation symmetric with respect to the polar axis? a. yes. b. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis. c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.

Answer

Explanation:

Step1: Test for polar - axis symmetry

Replace $\theta$ with $-\theta$ in the equation $r = 2\cos\theta$. We know that $\cos(-\theta)=\cos\theta$. So, when we substitute $\theta$ with $-\theta$, the equation becomes $r = 2\cos(-\theta)=2\cos\theta$, which is the original equation.

Step2: Conclusion for polar - axis symmetry

Since the equation remains the same after replacing $\theta$ with $-\theta$, the graph of the polar equation $r = 2\cos\theta$ is symmetric with respect to the polar axis.

Answer:

A. Yes.