Problem 17 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \sec 90^{\circ} $$

Step-by-Step Solution

Answer

The expression \(\sec(90^{\circ})\) is undefined.

Step 1: Find the cosine value

The first step is to find the cosine value of the given angle. In this case, \(\cos(90^{\circ}) = 0\). This is a known trigonometric value.

Step 2: Calculate the secant

The secant function is the reciprocal of the cosine function, so \(\sec(90^{\circ}) = \frac{1}{\cos(90^{\circ})}\)

Step 3: Determine if the expression is defined

If the denominator of a fraction is 0, then the expression is undefined. Because \(\cos(90^{\circ}) = 0\), taking the reciprocal results in an undefined value.

Key Concepts

Secant

The secant function, often represented as \( \sec \), is a less commonly used but important trigonometric function. It is the reciprocal of the cosine function. This means that to find secant's value for any angle, you take 1 divided by the cosine of that angle. This relationship can be written algebraically as:

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

This tells us that whenever cosine has a non-zero value, secant will have a defined value. However, since we cannot divide by zero, secant becomes undefined whenever cosine becomes zero.
The secant function is particularly useful in aspects of trigonometry where we deal with reciprocal relationships and often appears in calculus related to certain integration problems.
Understanding the behavior of secant, especially its undefined nature when cosine is zero, helps in graphing it or using it in calculations involving other trigonometric functions.

Undefined Expressions

In mathematics, an undefined expression is one that does not produce a valid or finite number. A common situation where this occurs is division by zero. When we have a fraction's denominator that equals zero, the fraction itself becomes undefined. This is crucial information for expressions involving reciprocal functions like secant.

• For example, if \( \sec(90^{\circ}) = \frac{1}{0} \), it means the expression is undefined.

The undefined nature occurs because there is no number that you can multiply by zero to get one, making it impossible to find a result.
Understanding undefined expressions is important because it helps avoid mistakes in calculations, such as attempting operations that lead to mathematical inaccuracies.

Cosine Function

The cosine function is one of the primary trigonometric functions, often written as \( \cos \). It represents the adjacent side over hypotenuse in a right-angled triangle, or the x-coordinate on the unit circle.

• \( \cos(\theta) \) gives values between -1 and 1 for angles measured in degrees or radians.

At particular angles, cosine takes on special values. For example, \( \cos(90^{\circ}) = 0 \). This is crucial in our problem as it directly affects the value of secant.
Cosine's value depends on the angle and is periodically repeated every 360 degrees (or \( 2\pi \) radians). Alongside sine, the cosine function is foundational in trigonometry and is heavily used in geometry, physics, and engineering.