To determine the symmetry of the polar graph given by the equation r = 4 cos(3θ), we need to check the conditions for symmetry about the x-axis, y-axis, and the origin.
1. Symmetry about the x-axis:
A polar graph is symmetric about the x-axis if replacing θ with -θ yields the same r value. Let’s test this:
Substituting -θ gives us:
r = 4 cos(3(-θ)) = 4 cos(-3θ) = 4 cos(3θ)
Since the value of r remains the same, this shows 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 gives the same r value. Let’s check:
Substituting π – θ gives:
r = 4 cos(3(π – θ)) = 4 cos(3π – 3θ) = 4 (-cos(3θ)) = -4 cos(3θ)
This changes the sign of r, indicating that the graph is not symmetric about the y-axis.
3. Symmetry about the origin:
A polar graph is symmetric about the origin if replacing θ with θ + π yields the same r value. Let’s test this condition:
Substituting θ + π gives:
r = 4 cos(3(θ + π)) = 4 cos(3θ + 3π) = 4 (-cos(3θ)) = -4 cos(3θ)
This also changes the sign of r, so the graph is not symmetric about the origin.
Conclusion:
Based on our analysis, the graph of r = 4 cos(3θ) is symmetric about the x-axis, but it is not symmetric about the y-axis or the origin.