Question

test for symmetry and graph the polar equation.
r = 5\cos(4\theta)
a. is the polar equation symmetrical with respect to the polar axis?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.
b. yes.
c. 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.
b. is the polar equation symmetrical with respect to the line \\( \theta=\frac{\pi}{2} \\)?
a. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line \\( \theta=\frac{\pi}{2} \\).
b. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the line \\( \theta=\frac{\pi}{2} \\).
c. yes.
c. is the polar equation symmetrical with respect to the pole?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the pole.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the pole.

Explanation:

• For symmetry about the polar axis (\(\theta = 0\)):
• Replace \(\theta\) with \(-\theta\). The cosine function has the property \(\cos(-x)=\cos(x)\). So, if \(r = 5\cos(4\theta)\), then \(r = 5\cos(- 4\theta)=5\cos(4\theta)\). A successful symmetry - test about the polar axis means the graph is symmetric about the polar axis.
• For symmetry about the line \(\theta=\frac{\pi}{2}\):
• Replace \((r,\theta)\) with \((-r,-\theta)\). We get \(-r = 5\cos(-4\theta)=5\cos(4\theta)\), or \(r=- 5\cos(4\theta)\), which is not the same as the original equation \(r = 5\cos(4\theta)\). A failed symmetry - test about the line \(\theta=\frac{\pi}{2}\) means the graph may or may not be symmetric about the line \(\theta=\frac{\pi}{2}\).
• For symmetry about the pole (\(r = 0\)):
• Replace \(r\) with \(-r\). We get \(-r=5\cos(4\theta)\), or \(r = - 5\cos(4\theta)\), which is not the same as the original equation \(r = 5\cos(4\theta)\). A failed symmetry - test about the pole means the graph may or may not be symmetric about the pole.

Answer:

a. B. Yes.
b. A. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line \(\theta=\frac{\pi}{2}\).
c. C. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the pole.