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The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It is commonly used in mathematics to describe oscillating movements, such as waves. The function is defined as the adjacent side over the hypotenuse in a right triangle. For the cosine function, values range between -1 and 1. This property forms the foundation for many applications, including calculating maximum values in trigonometric functions. The standard equation used for cosine is \[y = a \, \cos(bx + c) +d\]where

• \(a\) affects the amplitude,
• \(b\) affects the period,
• \(c\) shifts the graph horizontally,
• \(d\) shifts the graph vertically.

Understanding these transformations allows us to manipulate and deduce information about the graph, such as its maxima and minima.

The maximum value refers to the highest point a function can reach. For functions incorporating the cosine, the maximum is reached when the cosine part equals 1. This is because cosine can oscillate only between -1 and 1.
In the equation provided in the exercise, \(r = 6 + 12 \cos \theta\), the maximum value can be found by setting \(\cos \theta = 1\). This substitution results in\[r = 6 + 12 \times 1 = 18\]Thus, the maximum value of \(r\) is 18. It is crucial to understand that identifying the maximum value often involves substituting the trigonometric function with its extreme values. This helps us find the upper and lower bounds of the function in question, which is particularly useful when dealing with oscillating functions.

The zeros of a function are the points at which the value of the function equals zero. In mathematical terms, we set the equation equal to zero and solve for the variable. Finding the zeros helps in understanding where the function crosses the axis, which is an important aspect of function graphing and analysis.
For the exercise in question, we set the equation for \(r\) to zero: \[6 + 12 \cos \theta = 0\]Solving for \(\theta\), we find:\[12 \cos \theta = -6 \ \cos \theta = \frac{-6}{12} = -0.5\]The angles \(\theta = \arccos(-0.5)\) and \(\theta = 2\pi - \arccos(-0.5)\) correspond to angles that yield a cosine of -0.5. These angles are typically known to be \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\).
Finding zeros is an exercise in understanding the behavior of trigonometric functions, and it helps in wider applications such as solving complex equations and modeling periodic phenomena.