Problem 17 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\sec \left(-\frac{9 \pi}{2}\right)\)

Step-by-Step Solution

Answer

The value \( \sec \left(-\frac{9\pi}{2}\right) \) is undefined because the cosine of the angle is zero, leading to division by zero.

Step 1: Understand the Definition of Secant

The secant function, \( \sec(\theta) \), is defined as \( \frac{1}{\cos(\theta)} \). This means finding \( \sec \left(-\frac{9\pi}{2}\right) \) involves finding \( \frac{1}{\cos \left(-\frac{9\pi}{2}\right)} \). A value is undefined when the cosine value is zero.

Step 2: Simplify the Angle

The angle \( -\frac{9\pi}{2} \) goes beyond one full rotation (which is \( 2\pi \)). To simplify, add \( 2\pi \) repeatedly until you get an equivalent angle between \(-\pi\) and \(\pi\): \[ -\frac{9\pi}{2} + 4\pi = -\frac{\pi}{2} \] Thus, the angle simplifies to \( -\frac{\pi}{2} \).

Step 3: Evaluate the Equivalent Angle's Cosine

Now find \( \cos \left(-\frac{\pi}{2}\right) \). The angle \( -\frac{\pi}{2} \) is the angle on the negative y-axis in the unit circle. Here, the x-coordinate (cosine value) is 0.

Step 4: Determine Secant of the Angle

Since \( \cos \left(-\frac{\pi}{2}\right) = 0 \), the expression for secant becomes \( \sec \left(-\frac{\pi}{2}\right) = \frac{1}{0} \). Division by zero is undefined.

Key Concepts

Secant Function

The secant function, denoted as \(\sec(\theta)\), is an essential part of trigonometry. It relates directly to the cosine function. Here's the key idea about secant:

• It's defined as the reciprocal of cosine, specifically, \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
• This means that if the cosine of an angle is zero, the secant will be undefined because division by zero is not possible.

We often use this function to work with angles and triangles in different domains, particularly when needing to utilize relationships that involve the lengths of sides relative to an angle. Understanding when \(\sec(\theta)\) is undefined or defined requires examining \(\cos(\theta)\) first.

Angle Simplification

When working with trigonometric functions, and especially for angles well beyond a standard circle, angle simplification is a handy step.A full circle rotation in radians is \(2\pi\). If an angle such as \(-\frac{9\pi}{2}\) is given, it extends beyond a single rotation.

• To simplify, you add or subtract \(2\pi\) continually until the angle is within the primary interval of \(-\pi\) to \(\pi\).
• In our example, \(-\frac{9\pi}{2} + 4\pi = -\frac{\pi}{2}\), reducing the angle to a recognizable form.

This step ensures the angle is easy to interpret on the unit circle, where functions like cosine and sine have known values.

Unit Circle Concept

The unit circle is a powerful tool for understanding trigonometric functions.It is a circle with a radius of one unit centered at the origin of a coordinate system.

• Angles are measured from the positive x-axis, with counterclockwise being positive.
• Every point on the unit circle corresponds to \((\cos(\theta), \sin(\theta))\), where \(\cos(\theta)\) is the x-coordinate.

For \(\theta = -\frac{\pi}{2}\), you reach the point \(\left(0, -1\right)\) on the negative y-axis.

• The x-coordinate, \(0\), is the value of \(\cos(\theta)\), meaning \(\sec(-\frac{\pi}{2})\) is undefined.

Using the unit circle, students can visualize and predict the behavior of trigonometric functions at major angle benchmarks and associated transformations.