To determine the symmetry of the polar graph given by the equation r = 8 cos(3θ), we need to analyze it based on the characteristics of polar coordinates.
1. Symmetry about the x-axis: A polar graph is symmetric about the x-axis if replacing θ with -θ results in the same equation. Let’s check this:
Substituting -θ into the equation:
- r = 8 cos(3(-θ)) = 8 cos(-3θ)
- Since cos(-x) = cos(x), we have:
- r = 8 cos(3θ)
This indicates that the graph is symmetric 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 produces the same equation. We check this next:
Substituting π – θ into the equation:
- r = 8 cos(3(π – θ)) = 8 cos(3π – 3θ)
- Using the property cos(α + π) = -cos(α), we can rewrite this as:
- r = -8 cos(3θ)
This result is not identical to the original r = 8 cos(3θ), so the graph is not symmetric about the y-axis.
3. Symmetry about the origin: A polar graph is symmetric about the origin if replacing r with -r and θ with θ + π gives the same equation. Let’s verify this:
Replacing r with -r and θ with θ + π in the original equation:
- -r = 8 cos(3(θ + π)) = 8 cos(3θ + 3π) = 8(-cos(3θ))
- Thus, we get:
- -r = -8 cos(3θ)
- Which simplifies to:
- r = 8 cos(3θ)
This shows that the graph is also symmetric about the origin.
Final Conclusion: The graph of r = 8 cos(3θ) is symmetric about the x-axis and the origin, but not symmetric about the y-axis.