Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \sec \frac{\pi}{3} $$

Step-by-Step Solution

Answer

\( \sec \frac{\pi}{3} = 2 \)

Step 1: Recognize the Relationship Between Trigonometric Functions

The secant function is the reciprocal of the cosine function. Therefore, to find \( \sec \frac{\pi}{3} \), we first need to identify \( \cos \frac{\pi}{3} \). Since \( \sec x = \frac{1}{\cos x} \), \( \sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} \).

Step 2: Determine the Value of \( \cos \frac{\pi}{3} \)

The angle \( \frac{\pi}{3} \) radians corresponds to \( 60^\circ \). From trigonometric values, we know that \( \cos 60^\circ = \frac{1}{2} \). Therefore, \( \cos \frac{\pi}{3} = \frac{1}{2} \).

Step 3: Calculate \( \sec \frac{\pi}{3} \)

Using the reciprocal identity, we find \( \sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} \). Substituting the value from Step 2, we get \( \sec \frac{\pi}{3} = \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Secant Function

The secant function, often abbreviated as "sec," is one of the six fundamental trigonometric functions. It is not as commonly used as sine or cosine, but it holds its own significant role. Simply put, the secant function is the reciprocal of the cosine function. This means:

• if you have an angle \( x \), then \( \sec x = \frac{1}{\cos x} \).
• This relationship implies that to determine the value of secant, you first need to know the value of cosine.

To better understand its application, consider the example of finding \( \sec \frac{\pi}{3} \). Since the secant function is the reciprocal of the cosine for a given angle, it's crucial to first determine \( \cos \frac{\pi}{3} \). This solid link between secant and cosine allows for easier calculations and problem-solving strategies. Whenever you come across the secant function, remember that it always leans on understanding the cosine value first.

Reciprocal Identity

The reciprocal identity is a key concept in trigonometry that helps us relate different trigonometric functions. When we say two functions are reciprocals, it means they multiply to give 1. The reciprocal identity for secant and cosine is:

• \( \sec x = \frac{1}{\cos x} \)
  This identity is extremely helpful because it allows us to calculate the secant function by first finding the cosine value. It simplifies many trigonometric problems because you can always flip the cosine to get secant.

In practical terms, whenever you see an angle \( x \) and need to find \( \sec x \), think about the reciprocal identity. First, find \( \cos x \). Once you have that, use its reciprocal to get \( \sec x \). This identity highlights an essential aspect of how trigonometric functions interact with each other, making calculations more systematic and straightforward.

Trigonometric Values

Trigonometric values are fundamental in calculating specific ratios for standard angles. These values are derived from the unit circle, where each angle corresponds to a specific point on the circle leading to precise cosine, sine, and other trigonometric function values.

• For example, when dealing with the angle \( \frac{\pi}{3} \), it aligns with \( 60^\circ \) on the unit circle.
• The value of \( \cos \frac{\pi}{3} \) is known to be \( \frac{1}{2} \).

Understanding these standard trigonometric values is crucial because they serve as the building blocks for solving many problems. For instance, knowing that \( \cos 60^\circ \) equals \( \frac{1}{2} \) directly helps you find \( \sec \frac{\pi}{3} \) using reciprocal identities. Becoming familiar with these values and the angles associated with them can make tackling trigonometry problems much easier and faster. Always remember that these values stem from geometric relationships on the unit circle itself.