The given polar equation is r = 5 cos(θ). To determine the symmetry of the graph, we can check for symmetry about different axes and the origin.
1. Symmetry About the X-Axis:
For symmetry about the x-axis, we replace θ with -θ. If the equation remains unchanged, then the graph has x-axis symmetry. Substituting -θ gives us:
r = 5 cos(-θ) = 5 cos(θ). Since the equation remains unchanged, the graph is symmetric about the x-axis.
2. Symmetry About the Y-Axis:
To check for symmetry about the y-axis, we replace θ with π – θ. So we can see if:
r = 5 cos(π – θ) = -5 cos(θ), which does not match the original equation. Hence, the graph is not symmetric about the y-axis.
3. Symmetry About the Origin:
To check for symmetry about the origin, we replace (r, θ) with (-r, θ + π). We can rewrite the equation as:
-r = 5 cos(θ + π). This simplifies to:
-r = 5 (-cos(θ)) = -5 cos(θ), which leads back to r = 5 cos(θ). Thus, the graph is also symmetric about the origin.
Conclusion:
The graph of the polar equation r = 5 cos(θ) is symmetric about the x-axis and the origin, but not about the y-axis.