Problem 22 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \sec \pi $$

Step-by-Step Solution

Answer

The value of \( \sec \pi \) is -1.

Step 1: Understanding the secant function

The secant function, denoted \(\sec x\), is the reciprocal of the cosine function. Thus, we have \(\sec x = 1/ \cos x\). So, to find the secant of an angle, we first need to find the cosine of that angle.

Step 2: Find the cosine of the given angle

The given angle in the exercise is \(\pi\) radians. The cosine of \(\pi\) radians is -1.

Step 3: Find the secant of the given angle

\(\sec \pi = 1/ \cos \pi = 1/ -1 = -1\).

Key Concepts

Secant Function

The secant function is a vital part of trigonometry. It's represented as \( \sec x \) and plays a crucial role in connecting the angle to the cosine function. The secant function is defined as the reciprocal of the cosine function. This means that \( \sec x = \frac{1}{\cos x} \). The cosine value at a specific angle gives us a way to calculate the secant. By simply taking the reciprocal of that cosine value, you obtain the secant value.

For instance, if \( \cos x = a \), then \( \sec x = \frac{1}{a} \). Therefore, whenever you need to find the secant of an angle, your first task is to determine the cosine of that angle. Once you have that, computing the secant becomes straightforward. The secant function is especially useful in various applications, such as in trigonometric identities and equations.

Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, radians offer a more natural measure for angles, especially in mathematical contexts. The idea is simple: the angle in radians is the length of the arc divided by the radius of the circle.

A full circle is comprised of \( 2\pi \) radians, which is equivalent to 360 degrees. Therefore, \( \pi \) radians correspond to half a circle or 180 degrees. This conversion is useful when you switch between working with degrees and radians. In trigonometry, radians provide a seamless way to integrate calculations involving arc lengths and angular velocities.

When dealing with trigonometric functions like secant in terms of radians, their properties often become more apparent, making calculations easier to interpret and visualize.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions add depth to the study of trigonometric relations. These include secant (\( \sec x \)), cosecant (\( \csc x \)), and cotangent (\( \cot x \)) functions. Each of these is the reciprocal of the primary trigonometric functions—cosine, sine, and tangent, respectively.

• Secant: This is the reciprocal of cosine, expressed as \( \sec x = \frac{1}{\cos x} \).
• Cosecant: Being the reciprocal of sine, it is noted as \( \csc x = \frac{1}{\sin x} \).
• Cotangent: It serves as the reciprocal of tangent, written as \( \cot x = \frac{1}{\tan x} \).

These functions are critical in solving trigonometric equations where inverse relationships are involved. They have specific graph characteristics, domain, and range, different from their primary counterparts. Understanding reciprocal trigonometric functions allows one to explore more complex trigonometric identities and functions with depth and insight.