Problem 9 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \sec 45^{\circ} $$

Step-by-Step Solution

Answer

\(\sec 45^{\circ} = \sqrt{2}\)

Step 1: Understand the secant function

The secant function is the reciprocal of the cosine function. It can be expressed as \( \sec \theta = \frac{1}{\cos \theta} \). In this case, \( \theta = 45^{\circ} \). So the task is to find the value of \( \sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} \)

Step 2: Calculate the cosine value

The cosine of 45 degrees is \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)

Step 3: Calculate the secant value

Substitute the calculated cosine value into the secant expression: \( \sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \)

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions. Its notation is \( \sec \theta \), where \( \theta \) represents the angle for which we are determining its trigonometric value. What's intriguing about the secant function is that it is directly linked to the cosine function. Specifically, the secant function is defined as the reciprocal of the cosine function. This relationship can be expressed mathematically as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding the concept of the secant function aids in analyzing periodic phenomena and solving geometric problems related to circles and triangles. In the context of the exercise, we needed to find \( \sec 45^{\circ} \). By taking the reciprocal of the cosine value, which we can determine using the unit circle or trigonometric tables, the task becomes manageable.

Cosine Function

The cosine function, often expressed as \( \cos \theta \), is another key member of the trigonometric function family. It plays a crucial role in describing the relationship between a right-angle triangle's adjacent side and its hypotenuse. Defined as the ratio of these two sides, this value reveals vital information about the triangle's angles. At \( 45^{\circ} \), a special angle in trigonometry, the cosine value is particularly interesting:

• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)

This result emerges from the geometry of an isosceles right triangle, where both non-hypotenuse sides are equal. By remembering these standard trigonometric values, calculations involving these angles become straightforward and efficient. Such understanding simplifies the process of finding the secant of \( 45^{\circ} \), as demonstrated in the exercise.

Reciprocal Function

In mathematics, a reciprocal function provides a unique approach to understanding and manipulating equations. The reciprocal of a number is simply 1 divided by that number. For functions like trigonometric functions, taking the reciprocal can unearth new insights and make complex calculations accessible. The significance of a reciprocal is observed in several trigonometric functions, including secant, cosecant \(( \text{csc} )\), and cotangent \(( \text{cot} )\). With secant, for instance, the reciprocal of the cosine function, an originally intricate angle calculation converts to:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Knowing and using reciprocal relationships, like \( \sec 45^{\circ} = \sqrt{2} \), enables students to confidently tackle trigonometric problems. It also lays the groundwork for deeper geometric and algebraic explorations, linked to the curiously symmetric behaviors of angles and their trigonometric functions across the coordinate plane.