To determine the symmetry of the graph described by the polar equation r = 3 cos(5θ), we need to analyze the properties of polar equations.
1. **Symmetry about the x-axis:** A polar graph is symmetric about the x-axis if replacing θ with -θ yields the same equation. Let’s test this:
When we replace θ with -θ:
r = 3 cos(5(-θ)) = 3 cos(-5θ) = 3 cos(5θ).
This shows that the equation remains the same, indicating symmetry about the x-axis.
2. **Symmetry about the y-axis:** A polar graph is symmetric about the y-axis if replacing θ with π – θ gives the same equation. Testing this:
When we replace θ with π – θ:
r = 3 cos(5(π – θ)) = 3 cos(5π – 5θ) = 3(-cos(5θ)) = -3 cos(5θ).
This indicates that we do not have symmetry about the y-axis since r does not equal the same expression as in our original equation.
3. **Symmetry about the origin:** A polar graph is symmetric about the origin if replacing r with -r and θ with θ + π yields the same equation. Testing this gives:
When we replace r with -r and θ with θ + π:
-r = 3 cos(5(θ + π)) = 3 cos(5θ + 5π) = 3 cos(5θ).
Again, while -r has changed its sign, we can see this does not suggest symmetry about the origin either.
In conclusion, the graph of r = 3 cos(5θ) is symmetric only about the x-axis.