Problem 36 - Precalculus: A Concise Course

Question

Evaluate the trigonometric function using its period as an aid. $$ \cos 3 \pi $$

Step-by-Step Solution

Answer

So, \(\cos 3 \pi = -1\).

Step 1: Identifying the period of cosine

The cosine function has a period of \(2\pi\). This means that every \(2\pi\), the function completes one full cycle and starts a new one. Therefore, \(\cos x = \cos (x + 2\pi)\) for all values of x.

Step 2: Applying the period property

Since we are tasked with finding \(\cos 3 \pi\), and we know that cosine has its period at \(2\pi\), therefore we can subtract \(2\pi\) from \(3\pi\) to get \(\pi\). This is because \(\cos 3 \pi = \cos (3 \pi - 2 \pi) = \cos \pi\).

Step 3: Evaluating the function

Now let's calculate \(\cos \pi\). \(\cos \pi = -1\).

Key Concepts

Period of Cosine

The period of a trigonometric function refers to the length of the interval after which the function starts to repeat its values. For the cosine function, this period is \(2\pi\). That is, \(\cos(x) = \cos(x + k\cdot2\pi)\) for any integer \(k\). This property is immensely useful in simplifying complex trigonometric calculations.

Understanding the period can be pivotal when evaluating trigonometric functions. Let's see this with an example: if you need to find the value of \(\cos(3\pi)\), you can apply the periodicity by recognizing that \(3\pi\) is \(\pi\) more than \(2\pi\), so \(\cos(3\pi)\) is the same as \(\cos(\pi)\), because they are one complete period apart. This insight allows for the evaluation of trigonometric functions at simpler, equivalent angles.

Evaluating Trigonometric Functions

Evaluating trigonometric functions such as cosine involves finding the function's value at specific angles. This goes beyond mere substitution into the function's formula; it requires understanding the function's behavior, symmetry, and special angles.

When evaluating, one approach is to first consider if the given angle is an integral multiple of \(\pi/2\) or \(\pi\), since cosine takes on familiar values at these angles, such as 1, 0, or -1. For angles not immediately recognized, techniques like using the function's period come into play. In the provided exercise, by recognizing that \(\cos(3\pi) = \cos(\pi)\), we've employed the period property to simplify the problem to evaluating \(\cos(\pi)\), which is a well-known value of -1.

Cosine Function Properties

The cosine function possesses several key properties that make it both predictable and easier to work with. It's an even function, meaning that \(\cos(-x) = \cos(x)\), and it showcases symmetry about the y-axis. Also, the cosine function has a maximum value of 1 and a minimum value of -1, corresponding to the amplitude of the function.

Other properties include its aforementioned periodicity and the fact that it’s a continuous function. These characteristics facilitate the thorough understanding and evaluation of cosine in various contexts. For instance, knowing the maximum and minimum values helps to quickly evaluate \(\cos(\pi)\) as -1 without further calculations. The interplay of these properties is what makes studying the cosine function, and trigonometry in general, a cornerstone of mathematics education.