Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 3 cos 5θ?

Solution:

r = 3 cos 5θ is the given polar equation. Let us test the symmetry.

1. Symmetric about the x-axis

If (r, θ) lies on the graph, then (r, -θ) or (-r, π - θ) lies on the graph.

2. Symmetric about the y-axis

If (r, θ) lies on the graph, then (r, -θ) or (-r, -θ) lies on the graph.

3. Symmetric about the origin

If (r, θ) lies on the graph, then (r, -θ) or (-r, π + θ) lies on the graph.

1) Now check the equation for (r, -θ)

r = 2 cos (-5θ) = 2 cos (5θ) = r

So the graph is symmetric about the x-axis.

2) Now check the equation for (-r, -θ)

- r = 2 cos (-5θ) = 2 cos (5θ) ⇒ r ≠ - r

So the graph is not symmetric about the y-axis.

3) Now check the equation for (-r, θ)

- r = 2 cos (5θ) ⇒ r ≠ - r

So the graph is not symmetric about the origin.

Therefore, the graph is symmetric about the x-axis.

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 3 cos 5θ?

Summary:

The graph is symmetric about the x-axis.