Problem 20 - Precalculus

Question

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions. $$\cos \left(-\frac{5 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The value of \( \cos \left(-\frac{5 \pi}{3}\right) \) is \( \frac{1}{2} \).

Step 1: Understanding Properties of Cosine

The cosine function is an even function. This means that \( \cos(-x) = \cos(x) \). Thus, \( \cos \left(-\frac{5\pi}{3}\right) = \cos \left(\frac{5\pi}{3}\right) \).

Step 2: Finding the Reference Angle

Since \( \frac{5\pi}{3} \) is more than \( 2\pi \), we need to find the equivalent angle within the range \( [0, 2\pi) \). Subtract \( 2\pi \) to get \( \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \). Since we want a positive angle, add \( 2\pi \) to \( -\frac{\pi}{3} \) to find the equivalent angle within one rotation: \( 2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3} \).

Step 3: Using Unit Circle Values

The reference angle for \( \frac{5\pi}{3} \) is \( \frac{\pi}{3} \). On the unit circle, \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Since cosine is an even function, \( \cos \left(\frac{5\pi}{3}\right) = \frac{1}{2} \).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry and is used to describe the position of a point along the x-axis on the unit circle. Imagine a circle with a radius of 1, situated at the origin of a coordinate plane. This circle is known as the unit circle.

• The angle in question determines a point on the circle, and the cosine of this angle corresponds to the x-coordinate of this point.
• For example, if you have an angle of \( \theta \), the cosine function, written as \( \cos(\theta) \), gives the horizontal distance from the center of the circle to where the terminal side of the angle intersects the circle.

By understanding this, you can easily find the cosine of various angles. For instance, using the unit circle's coordinates, you know that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). This property allows you to solve problems like the one in the exercise.

Even Function

Understanding even functions is crucial to solving trigonometric problems efficiently. An even function is symmetric with respect to the y-axis. This symmetry implies certain simplifications.

• For cosine, an even function, this means that \( \cos(-x) = \cos(x) \). So, the cosine value is the same for both a positive and its corresponding negative angle.
• For example, if you need to find \( \cos(-\frac{5\pi}{3}) \), you can directly equate it to \( \cos(\frac{5\pi}{3}) \) without any additional calculations.

This property is extremely handy when dealing with negative angles, as it greatly simplifies the solution process. In essence, it allows you to ignore negatives when dealing with the cosine function, focusing instead on finding the cosine of the positive angle.

Reference Angle

Reference angles are an essential concept in trigonometry, helping to simplify finding trigonometric values for angles that are not on the unit circle's basic positions.

• A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
• This is useful because the trigonometric functions of an angle and its reference angle will share certain values, differing only by sign depending on the angle's quadrant.

For instance, when working with \( \frac{5\pi}{3} \), it's beneficial to find its reference angle. Since \( \frac{5\pi}{3} \) is equivalent to one complete rotation plus an extra \( \frac{\pi}{3} \), we can find its reference angle to simplify calculations. Here, the reference angle is \( \frac{\pi}{3} \), where \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). This makes it significantly easier to determine the cosine of angles like \( \frac{5\pi}{3} \) by providing a known value to reference.