Determine if the graph of r = 4cos5θ is symmetric about the x-axis, the y-axis, or the origin.

Solution:

Function graph is symmetric about

1. the origin or the pole if f(r, θ) = f(-r,-θ)
2. the y-axis if f(r, θ) = f(-r,θ)
3. the x-axis, if f(r, θ) = f(r,-θ)

1) Let us test for symmetry about the x-axis

f(r,θ) --> r = 4 cos5θ

f(r,-θ) --> r = 4 cos5(-θ) 

r = 4 cos5θ

So the graph is symmetric about the x-axis

2) Let us test for symmetry about the y-axis

f(r,θ) --> r = 4 cos5θ

f(-r,θ) --> -r = 4 cos5θ

So the graph is not symmetric about the y-axis

3) Let us test for symmetry about the origin

f(r,θ) --> r = 4 cos5θ

f(-r,-θ) --> -r = 4 cos5(-θ) 

r = -4 cos5θ

So the graph is not symmetric about the origin

Therefore, we conclude that the graph is symmetric about the x-axis.

Determine if the graph of r = 4cos5θ is symmetric about the x-axis, the y-axis, or the origin.

Summary:

The graph of r = 4cos5θ is symmetric about the x-axis.