Problem 11 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\sec \frac{3 \pi}{2}\)

Step-by-Step Solution

Answer

\( \sec \frac{3\pi}{2} \) is undefined because \( \cos \frac{3\pi}{2} = 0 \).

Step 1: Understand the Secant Function

The secant function is the reciprocal of the cosine function. Thus, \( \sec \theta = \frac{1}{\cos \theta} \). We'll need to find \( \cos \frac{3\pi}{2} \) first to determine \( \sec \frac{3\pi}{2} \).

Step 2: Locate the Angle on the Unit Circle

The angle \( \frac{3\pi}{2} \) radians corresponds to \( 270^\circ \). This angle is located at the bottom of the unit circle along the negative y-axis, where the coordinates are (0, -1).

Step 3: Find the Cosine Value

The cosine of an angle is the x-coordinate of its corresponding point on the unit circle. Therefore, \( \cos \frac{3\pi}{2} = 0 \).

Step 4: Determine the Secant Value

Using \( \sec \theta = \frac{1}{\cos \theta} \), we try to find \( \sec \frac{3\pi}{2} \). However, since the cosine is zero, the expression \( \frac{1}{0} \) is undefined.

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It simplifies the process of finding values of trigonometric functions by mapping angles to coordinates in a circle with a radius of 1.
It's centered at the origin of the coordinate plane. Each point on the unit circle has coordinates

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

To locate a specific angle such as \( \frac{3\pi}{2} \) radians, we translate radians into a position on the unit circle.
The angle \( \frac{3\pi}{2} \) is equivalent to 270 degrees. This is positioned at the bottom of the circle on the negative y-axis. Therefore, its coordinates are (0, -1).
The unit circle helps easily determine trigonometric values like cosine and sine, crucial for finding secant functions.

Exploring Trigonometric Functions: Focus on Secant

Trigonometric functions are essential for analyzing mathematical angles and calculations. They include sine, cosine, tangent, cosecant, secant, and cotangent.
The secant function, noted as \( \sec \theta \), is particularly interesting because it is the reciprocal of the cosine function. The primary relationship is:

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

To compute \( \sec \frac{3\pi}{2} \), we first need \( \cos \frac{3\pi}{2} \). Using the unit circle, we know that at \( \frac{3\pi}{2} \) radians, the coordinates are (0, -1), so:

• The cosine value, or x-coordinate, is 0.

This makes \( \sec \frac{3\pi}{2} \) undefined because you can't divide by zero. The secant function, like its reciprocal basis cosine, plays a critical role in trigonometric analysis by establishing relationships between different quadrantal angles.

The Concept of Undefined Expressions

In mathematics, expressions become undefined when they involve operations that aren't valid within the number system. A common case is division by zero.
Any expression where a number divides by zero lacks a meaningful result, thus is termed undefined.
For example, in the case of the trigonometric secant function, \( \sec \theta = \frac{1}{\cos \theta} \) becomes undefined if \( \cos \theta = 0 \).

• This happens at specific parts of the unit circle where cosine values reach zero.
• These quadrants are key as they consistently cause expressions like \( \sec \theta \) and \( \cot \theta \) to be undefined..

Recognizing undefined cases is important in solving trigonometric problems, as they often indicate situations with restricted solutions. Learning these crucial points allows for more effective understanding and problem-solving in trigonometry.