Answer:
The graph is symmetric about the x-axis.
Step-by-step explanation:
1. Symmetry about the x-axis: If the point (r, θ ) lies on the graph, then the point (r, -θ ) or (-r, π - θ ) also lies on the graph.
2. Symmetry about the y-axis: If the point (r, θ ) lies on the graph, then the point (r, π - θ ) or (-r, -θ ) also lies on the graph.
3. Symmetry about the origin: If the point (r, θ ) lies on the graph, then the point (-r, θ ) or (r, π + θ ) also lies on the graph.
The given polar equation is
[tex]r=2\cos (5\theta)[/tex]
Check the equation by (r, -θ).
[tex]r=2\cos (-5\theta)=2\cos (5\theta)=r[/tex]
Therefore, the graph is symmetric about the x-axis.
Check the equation by (-r, -θ).
[tex]-r=2\cos (-5\theta)=2\cos (5\theta)=r\neq -r[/tex]
Therefore, the graph is not symmetric about the y-axis.
Check the equation by (-r, θ).
[tex]-r=2\cos (5\theta)=2\cos (5\theta)=r\neq -r[/tex]
Therefore, the graph is not symmetric about the origin.
