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When analyzing the symmetry of a curve in polar coordinates, identifying polar axis symmetry is a fundamental step. A curve has polar axis symmetry if replacing \(\theta\) with \( -\theta\) doesn't alter the original equation representing the curve.

Let's apply this to our exercise's equation \(r=5+4 \cos \theta\). By substituting \(\theta\) with \( -\theta\) and considering the even property of the cosine function—which states that \(\cos(-\theta) = \cos(\theta)\)—we observe that the curve remains unchanged. This means the graph of the equation \(r=5+4 \cos \theta\) is symmetrical with respect to the polar axis, a line that passes through the pole and is aligned with the vertical direction of the graph.

Line symmetry in the context of polar coordinates involves flipping the graph across a particular line, such as \(\theta = \frac{\pi}{2}\), to check if it maps onto itself. In our exercise, we are instructed to assess the symmetry with respect to the line \(\theta = \frac{\pi}{2}\).

To do this, substitute \(\pi - \theta\) for \(\theta\) in the given equation \(r=5+4 \cos \theta\). Thanks to the even nature of the cosine function, \(\cos(\pi - \theta)\) equals \(\cos(\theta)\). Therefore, the equation remains consistent, showing that it possesses line symmetry about \(\theta = \frac{\pi}{2}\). This particular line is a reference line in polar coordinate graphs, perpendicular to the polar axis.

Pole symmetry is related to a shape being symmetrical about the pole (origin) in polar coordinates. If substituting \(\theta + \pi\) for \(\theta\) in the polar equation results in the same or a reflected version (corresponding to \(r\) being negative) of the original equation, the graph has pole symmetry.

For our function \(r=5+4 \cos \theta\), replacing \(\theta\) with \(\theta + \pi\) yields \(r=5+4 \cos(\theta + \pi)\), which simplifies to \(r=5-4 \cos(\theta)\) due to the cosine function's property of producing a negative result when shifted by \(\pi\). The transformed equation doesn't match the original, indicating the graph doesn't have pole symmetry, a rare property for most polar graphs.

The concepts of even and odd functions play a pivotal role in determining the symmetry of polar equations. In trigonometry, a function \(f(\theta)\) is called even if \(f(-\theta) = f(\theta)\), and it's called odd if \(f(-\theta) = -f(\theta)\).

In our exercise, the cosine function is even, as demonstrated by \(\cos(-\theta) = \cos(\theta)\), which allows us to establish the presence of polar axis and line symmetry in the graph. The sine function, by contrast, is odd. Such properties greatly simplify the process of testing for various symmetries in polar curves, making it easier for students to grasp and apply these concepts.

Understanding the properties of the cosine function aids immensely in working with polar coordinates. One essential property is its periodicity: \(\cos(\theta + 2\pi) = \cos(\theta)\), which means that the function repeats its values over intervals of \(2\pi\). Another crucial property is that the cosine function is an even function: \(\cos(-\theta) = \cos(\theta)\).

These characteristics of the cosine function have been put into practice in our exercise to test for symmetry. We've relied on the fact that adding \(\pi\) to the argument results in the negative cosine value (\(\cos(\theta + \pi) = -\cos(\theta)\)), proving useful in evaluating pole symmetry. Furthermore, we've highlighted how these properties confirm the polar axis and line symmetries of the given equation \(r=5+4 \cos \theta\). As a result, students can remember that understanding trigonometric properties can greatly enhance their analysis of polar graphs.