Problem 27 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\cos \left(-\frac{7 \pi}{3}\right)$$

Step-by-Step Solution

Answer

\( \cos \left(-\frac{7 \pi}{3}\right) = \frac{1}{2} \).

Step 1: Understand the function

The given function is \( \cos \left(-\frac{7 \pi}{3}\right) \). The cosine function is periodic with a period of \(2\pi\). This means that we can add or subtract multiples of \(2\pi\) to simplify the angle.

Step 2: Simplify the angle

To simplify \( \cos \left(-\frac{7 \pi}{3}\right) \), we first reduce \( -\frac{7 \pi}{3} \) by adding \( 2\pi \) multiples until it falls within one complete circle (\(0\) to \(2\pi\)). Since \( 2\pi = \frac{6\pi}{3} \), add \(\frac{6\pi}{3}\) to get:\[ -\frac{7\pi}{3} + \frac{6\pi}{3} = -\frac{\pi}{3} \]Thus, \( \cos \left(-\frac{7 \pi}{3}\right) = \cos \left(-\frac{\pi}{3}\right) \).

Step 3: Use cosine even function property

Cosine is an even function, which means \( \cos(-x) = \cos(x) \). Therefore:\[ \cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) \]

Step 4: Evaluate cosine of \( \frac{\pi}{3} \)

From the unit circle or trigonometric values, we know that:\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, typically denoted as \( \cos(\theta) \). It relates an angle of a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse. This function varies as the angle \( \theta \) changes, mapping all angles to corresponding values between -1 and 1.

The cosine function is defined for all real numbers, and its graph is a smooth curve that oscillates between these two bounds. It has several important properties, such as being an even function and having a period of \(2\pi\). Cosine values are useful in many fields, including physics and engineering, for modeling wave behavior, among other applications.

Periodic Functions

Periodic functions have the property that they repeat their values at regular intervals. For the cosine function, this interval, known as the period, is \(2\pi\).

This means for any angle \( \theta \), the cosine function becomes:

• \( \cos(\theta + 2\pi) = \cos(\theta) \)
• \( \cos(\theta - 2\pi) = \cos(\theta) \)

This periodicity is what allows us to simplify angles in trigonometric functions. For example, if an angle is outside the range of \([0, 2\pi]\), we can "wrap" it within one complete circle by either adding or subtracting \(2\pi\) until it falls within this standard range. This property is particularly useful when evaluating trigonometric functions with angles obtained in various situations, such as rotations and oscillatory motions.

Unit Circle

The unit circle is a powerful tool in trigonometry, often used to define the cosine and sine functions for all angles. It is a circle centered at the origin of a coordinate plane with a radius of 1.

On the unit circle:

• Each point is defined by its coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
• The cosine value corresponds to the x-coordinate of these points.

The unit circle allows easy determination of cosine values for common angles like \( \frac{\pi}{3}, \frac{\pi}{2}, \pi \), among others, without needing to calculate ratios from triangles repeatedly. It essentially provides a visual and conceptual understanding of how cosine behaves as a function of angles.

Even Function Property

In mathematics, an even function is one that satisfies the condition \( f(-x) = f(x) \) for all values of \( x \) in its domain. The cosine function is a prime example of an even function.

Because cosine is even, we have:

• \( \cos(-\theta) = \cos(\theta) \)
• This property enables simplification of expressions involving negative angles.

For instance, the function \( \cos(-\frac{\pi}{3}) \) can be directly rewritten as \( \cos(\frac{\pi}{3}) \). This property of the cosine function helps us evaluate trigonometric expressions more easily, and it also reflects the symmetric nature of cosine with respect to the y-axis on the unit circle.