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To understand symmetry about the x-axis in polar coordinates, you want to observe if flipping the angle across the x-axis leaves the graph unchanged. In simpler terms, you achieve this by replacing \( \theta \) with \(-\theta \) in the polar equation and see if it still matches the original equation.

For example, in the equation \( r^2 = \cos(4\theta) \), we replace \( \theta \) with \(-\theta \). The result is \( r^2 = \cos(4(-\theta)) = \cos(-4\theta) = \cos(4\theta) \). This shows that the expression hasn’t changed, confirming x-axis symmetry.

Key points:

• Replace \( \theta \) with \(-\theta \).
• Check if the equation remains the same.
• If unchanged, the graph is symmetric about the x-axis.

By grasping this concept, determining x-axis symmetry becomes a simple substitution task.

Symmetry about the y-axis in polar graphs requires checking whether transforming the radius and angle leads back to the original equation. Specifically, replace \( \theta \) with \( \pi - \theta \). This idea is akin to reflecting across the y-axis.

In the given equation, \( r^2 = \cos(4\theta) \), applying this replacement, we get \( r^2 = \cos(4(\pi - \theta)) = \cos(4\pi - 4\theta) \). Using the identity \( \cos(2\pi k - x) = \cos(x) \), we deduce \( \cos(4\theta) = \cos(4\pi - 4\theta) \). This indicates that the graph maintains symmetry about the y-axis.

Key steps:

• Substitute \( \theta \) with \( \pi - \theta \).
• Use cosine properties to verify equivalence.
• If the equation remains unchanged, y-axis symmetry is present.

This method provides a clear pathway to verify y-axis symmetry with ease.

Determining symmetry about the origin in polar coordinates involves checking if adjusting both the radius and angle retains the original formula. Essentially, replace \((r, \theta)\) with \((-r, \theta + \pi)\), or equivalently, only \(\theta\) with \(\theta + \pi\).

Using \( r^2 = \cos(4\theta) \) as an example, substituting \( \theta \) with \( \theta + \pi \), the equation becomes \( r^2 = \cos(4(\theta + \pi)) = \cos(4\theta + 4\pi) \). The cosine property \( \cos(x + 2\pi) = \cos(x) \) confirms \( \cos(4\theta) = \cos(4\theta + 4\pi) \), thus indicating origin symmetry.

Important points:

• Replace \( \theta \) with \( \theta + \pi \).
• Check if the expression is congruent to the original.
• If unchanged, the graph is symmetric about the origin.

This approach simplifies checking origin symmetry for any polar equation, helping you quickly test and verify your results.