To determine the symmetry of the graph given by the polar equation r = 5 cos(3θ), we can analyze it based on the definitions of symmetry in polar coordinates.
1. Symmetry about the x-axis:
A graph is symmetric about the x-axis if replacing θ with -θ yields the same result for r. Let’s test this:
r = 5 cos(3(-θ))
= 5 cos(-3θ)
= 5 cos(3θ)
Since cos(-x) = cos(x), we find that replacing θ with -θ gives us the original equation. Therefore, the graph is symmetric about the x-axis.
2. Symmetry about the y-axis:
A graph is symmetric about the y-axis if replacing θ with π – θ gives the same r. Let’s check this condition:
r = 5 cos(3(π - θ))
= 5 cos(3π - 3θ)
= 5(-cos(3θ))
Since this does not equal the original equation in terms of a positive r (we have a negative), the graph does not exhibit symmetry about the y-axis.
3. Symmetry about the origin:
A graph is symmetric about the origin if replacing (r, θ) with (-r, θ + π) yields the same equation:
-r = 5 cos(3(θ + π))
= 5 cos(3θ + 3π)
= 5(-cos(3θ))
This leads us to:
-r = -5 cos(3θ) r = 5 cos(3θ)
Since this is indeed the original equation, the graph is symmetric about the origin as well.
Conclusion:
The graph of the equation r = 5 cos(3θ) is symmetric about the x-axis and the origin, but not about the y-axis.