Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 4 cos 3θ. me solve rn i need this, also include the proving part.

Symmetry can be determined visually in a graph. If both graphs are mirror-image of each other, then both of the equations are symmetric. But, you can also determine it analytically through testing the symmetry. These are the rules:

If
f(r, θ) = f(r,-θ), symmetric to the polar axis or the x-axis
f(r, θ) = f(-r,θ), symmetric to the y-axis
f(r, θ) = f(-r,-θ), symmetric to the pole or the origin

Test for symmetry about the x-axis
f(r,θ): r=4 cos3θ
f(r,-θ): r = 4 cos3(-θ) ⇒ r = 4 cos3θ
∴The graph is symmetric about the x-axis.

Test for symmetry about the y-axis
f(r,θ): r=4 cos3θ
f(-r,θ): -r = 4 cos3θ
∴The graph is not symmetric about the y-axis.

Test for symmetry about the origin
f(r,θ): r=4 cos3θ
f(-r,-θ): -r = 4 cos3(-θ) ⇒ r = -4 cos3θ
∴The graph is not symmetric about the origin.

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