algebraically identify the symmetries of r^2=4cos2theta

To algebraically identify the symmetries of the equation r^2 = 4cos(2θ), we can use the properties of the cosine function and the polar coordinate system.

Let's first rewrite the equation using the double-angle identity for cosine:

r^2 = 4cos^2(θ) - 4sin^2(θ)

Next, we simplify using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1:

r^2 = 4cos^2(θ) - 4(1 - cos^2(θ))

Simplifying further:

r^2 = 4cos^2(θ) - 4 + 4cos^2(θ)

r^2 = 8cos^2(θ) - 4

From this equation, we can see that r^2 depends only on the value of cos^2(θ). Since cosine is an even function, meaning cos(-θ) = cos(θ), we can conclude that cos^2(-θ) = cos^2(θ).

Therefore, the equation r^2 = 8cos^2(θ) - 4 possesses symmetry with respect to the polar axis (θ reflected to -θ) because the value of r^2 remains the same.

In simple terms, the equation has a symmetry where the graph remains unchanged when reflected across the x-axis.

Additionally, the equation does not have symmetry with respect to the pole (r reflected to -r). This is because changing the sign of r would result in changing the value of r^2.

Hence, the algebraic identification of the symmetries of the equation r^2 = 4cos(2θ) is symmetry with respect to the polar axis (x-axis), but no symmetry with respect to the pole (origin).