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Polar equations, such as the one in our exercise, describe a relationship between the radius \( r \) and the angle \( \theta \) in the polar coordinate system. In this system, each point on the plane is determined by \( r \), the distance from the origin, and \( \theta \), the angle measured from the positive x-axis to the line segment connecting the origin with the point.

The equation \( r=4 \cos 2 \theta \) uses the cosine function to modulate the radius as the angle changes. This ties the polar equation directly to the shape of the graph, which often results in elegant curves and loops. Interpreting polar equations requires thinking in a circular rather than rectangular grid, which can be challenging but also rewarding as it provides a new perspective on graphing.

Creating a table of values is an essential step in plotting the graph of a polar equation. A table of values lists pairs of radius and angle measurements. For the equation \( r=4 \cos 2 \theta \), we calculate corresponding \( r \) values for a variety of \( \theta \) values. It's insightful to use a systematic approach when choosing the angles, often starting from 0 and increasing in regular increments, like \( \frac{\pi}{8} \).

By substituting these \( \theta \) values into the polar equation, we get a set of \( r \) values, which we then plot on the polar grid. This process allows us to capture the most significant points of the graph, revealing the curve's overall shape. The table of values serves as a blueprint for the resultant graph.

Radians are a unit of angular measure used in mathematics to describe angles by the length of the arc they subtend on a unit circle. While degrees are more familiar in everyday use, radians offer a natural way to express angles when dealing with trigonometric functions and calculus.

One radian is the angle made when the arc's length is equal to the radius of the circle. There are \( 2\pi \) radians in a full circle, so a radian can be considered to be approximately 57.3 degrees. The choice of radians in plotting the polar equation is critical because trigonometric functions like the cosine, inherent to our exercise, use radians as their natural input.

The cosine function is one of the fundamental trigonometric functions, describing the ratio of the adjacent side to the hypotenuse in a right-angled triangle. But beyond triangles, it also has an essential role in waveforms and circular motion and, as such, is a significant component in polar equations.

In our exercise, the function appears as \( 4 \cos 2 \theta \) in the polar equation. The number 4 scales the amplitude of the cosine wave, and the factor of 2 inside the cosine function affects the frequency, indicating the graph will complete two full cycles of oscillation between 0 and \( 2\pi \) radians. Plotting this function in polar coordinates gives us a rose curve with certain symmetries and periodicity characteristic of the cosine function.