Question

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = 4 cos 5θ

Answer 1

To check symmetry with respect to the x-axis, see if r(-θ) is the same as r(θ).

Since the cosine is an even function

r(-θ) = 4 cos(5(-θ)) = 4 cos(-5θ) = 4 cos(5θ) = r(θ).

It is symmetric with respect to the x-axis.

To check symmetry with respect to the y-axis, see if r(π - θ) is the same as r(θ).

Using the difference of angles formulas for the cosine

r(π - θ) = 4 cos(5(π - θ)) = 4 cos(5π - 5θ) = 4 cos(5π) cos(5θ) - 4 sin(5π) sin(5θ)

= -4 cos(5θ) ≠ r(θ)

This is not the same, so the graph is not symmetric with respect to the y-axis.

For origin symmetry, the condition would be r(-θ ) = - r(θ). From the first analysis, we see that this is not the case. So the graph is not symmetric with respect to the origin. (You can also check for origin symmetry by checking to see if r(θ + π) = r(θ).)