Problem 6 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

Undefined, as division by zero is not possible.

Step 1: Understanding Secant

First, recall that the secant function is the reciprocal of the cosine function: \[ \sec(x) = \frac{1}{\cos(x)} \] So we need to find \( \cos \left( \frac{\pi}{2} \right) \) to compute \( \sec \left( \frac{\pi}{2} \right) \).

Step 2: Value of Cosine at \( \frac{\pi}{2} \)

The cosine of \( \frac{\pi}{2} \) radians (or 90 degrees) is 0. This is a standard angle on the unit circle where the point is (0, 1), and the x-coordinate, which gives the cosine value, is 0. \[ \cos \left( \frac{\pi}{2} \right) = 0 \]

Step 3: Calculate Secant Value

Using the reciprocal identity of secant, we have: \[ \sec \left( \frac{\pi}{2} \right) = \frac{1}{\cos \left( \frac{\pi}{2} \right)} = \frac{1}{0} \] Since division by zero is undefined in mathematics, \( \sec \left( \frac{\pi}{2} \right) \) is undefined.

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane. Imagine drawing a circle on a piece of graph paper with its center at the point (0, 0). As you move around the circle, you can measure angles from the positive x-axis, moving counterclockwise.

On the unit circle:

• Cosine (cos) of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.
• Sine (sin) of an angle is the y-coordinate of that point.
• At specific standard angles, like 0, \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \), the coordinates are well-known and used frequently.

For \( \frac{\pi}{2} \) (which is 90 degrees), the coordinate on the unit circle is (0, 1). Thus, the cosine at this angle is 0 because the x-coordinate of this point is 0. Understanding which points correspond to standard angles on the unit circle helps simplify problem-solving in trigonometry.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions extend basic trigonometric functions by taking their reciprocals. The most common ones include:

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

Understanding these relationships is critical because when the original function (like cosine) is zero or undefined, its reciprocal will be undefined as well. This occurs because dividing by zero is mathematically impossible. In the exercise, when you find \( \sec \left( \frac{\pi}{2} \right) \), you're looking for the reciprocal of 0, which is undefined.

Undefined Values in Trigonometry

In trigonometry, undefined values occur whenever a function's evaluated result requires a division by zero. This commonly happens with reciprocal trigonometric functions. For instance:

• If the cosine of an angle is 0, then the secant of that angle is undefined. This is because \( \sec(x) = \frac{1}{\cos(x)} \), and \( \frac{1}{0} \) is not possible.
• Similarly, if the sine of an angle is 0, the cosecant will be undefined.
• If tangent is undefined at a point, its reciprocal, cotangent, will be zero, due to the fact that \( \cot(x) = \frac{1}{\tan(x)} \).

These undefined values help point out key characteristics and restrictions of trigonometric functions. Recognizing where these occur on the unit circle can make it easier to predict and understand the behavior of trig functions across different angles.