Problem 339 - Precalculus

Question

For the following exercises, find the exact value of the given expression. $$ \sec \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The exact value of \(\sec \frac{\pi}{4}\) is \(\sqrt{2}\).

Step 1: Understand the Secant Function

The secant function, denoted as \(\sec(x)\), is the reciprocal of the cosine function. Therefore, \(\sec(x) = \frac{1}{\cos(x)}\). To find \(\sec \frac{\pi}{4}\), we first need to find \(\cos \frac{\pi}{4}\) and then take its reciprocal.

Step 2: Determine the Cosine of \(\frac{\pi}{4}\)

The angle \(\frac{\pi}{4}\) radians is equivalent to 45 degrees. We know that \(\cos(45^\circ) = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

Step 3: Calculate the Secant of \(\frac{\pi}{4}\)

Since \(\sec(x) = \frac{1}{\cos(x)}\), we have \(\sec \frac{\pi}{4} = \frac{1}{\cos \frac{\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}}\).

Step 4: Simplify the Expression

To simplify \(\frac{1}{\frac{\sqrt{2}}{2}}\), multiply the numerator and the denominator by 2 to get \(\frac{2}{\sqrt{2}}\). Further simplifying by multiplying by \(\frac{\sqrt{2}}{\sqrt{2}}\) gives \(\frac{2\sqrt{2}}{2} = \sqrt{2}\).

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions, and it's often represented as \( \sec(x) \). What makes the secant unique is that it is the reciprocal of the cosine function. This means that if you know the cosine of an angle, you can easily find the secant by flipping it. For any angle \( x \), the secant can be calculated as:\[ \sec(x) = \frac{1}{\cos(x)} \]A handy aspect of the secant is that it helps us understand other reciprocal trigonometric functions, like cosecant and cotangent. For example, if you know that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), then finding \( \sec(\frac{\pi}{4}) \) is straightforward:- First, find the cosine value: \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).- Then, take the reciprocal: \( \sec(\frac{\pi}{4}) = \frac{1}{\frac{\sqrt{2}}{2}} \).Secant is frequently used in solving trig equations and in calculus, especially when dealing with trigonometric identities and integrals.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the standard trigonometric functions: sine, cosine, and tangent by inverting or taking the reciprocal of these functions.

• Secant (\( \sec \)): This is the reciprocal of cosine. Defined as \( \sec(x) = \frac{1}{\cos(x)} \).
• Cosecant (\( \csc \)): This is the reciprocal of sine, \( \csc(x) = \frac{1}{\sin(x)} \).
• Cotangent (\( \cot \)): This is the reciprocal of tangent, \( \cot(x) = \frac{1}{\tan(x)} \).

These functions are particularly useful in trigonometry and calculus because they help simplify the computation of angles and the evaluation of trigonometric equations. Knowing the reciprocal functions can also aid in transforming complex trigonometric expressions into simpler forms, which are easier to solve or integrate. For example, if you know \( \csc(x) \) and \( \sec(x) \), you can quickly determine values and relationships without recalculating the original sine or cosine values. In this way, reciprocal functions expand your mathematical toolkit for tackling various problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are fundamental in simplifying trigonometric expressions and solving equations. Here are a few key identities to keep in mind:

• Pythagorean Identities: These relate angles and provide foundational truths: \[ \sin^2(x) + \cos^2(x) = 1 \]
• Reciprocal Identities: These show how functions relate to their reciprocals: \[ \sec(x) = \frac{1}{\cos(x)}, \quad \csc(x) = \frac{1}{\sin(x)}, \quad \cot(x) = \frac{1}{\tan(x)} \]
• Angle Sum and Difference Identities: These are useful for finding the sine, cosine or tangent of the sum or difference of two angles.

Understanding these identities is crucial for manipulating and simplifying expressions in a variety of mathematical contexts, from geometry to calculus. When dealing with complex trigonometric expressions, these identities can turn complicated problems into manageable ones and are essential tools for any mathematics student.