Problem 105 - Calculus for Biology and Medicine

Question

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Step-by-Step Solution

Answer

The domain of \( y = \sec x \) is all real numbers except odd multiples of \( \frac{\pi}{2} \).

Step 1: Understand the reciprocal relationship

The function \( y = \sec x \) is defined as the reciprocal of the cosine function, \( \sec x = \frac{1}{\cos x} \). This implies that wherever \( \cos x \) is zero, \( \sec x \) will be undefined.

Step 2: Identify where \(\cos x\) is zero

The cosine function, \( \cos x \), equals zero at odd multiples of \( \frac{\pi}{2} \). This can be written as \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer (e.g., \( k = 0, \pm1, \pm2, \ldots \)).

Step 3: Determine the domain of \(\sec x\)

Since \( \sec x \) is the reciprocal of \( \cos x \), \( \sec x \) will be undefined wherever \( \cos x = 0 \). Therefore, \( \sec x \) is undefined at odd multiples of \( \frac{\pi}{2} \), and its domain is all real numbers except at these points.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This relationship means that wherever the cosine function is defined, the secant function can also be defined, except where the cosine equals zero because division by zero is undefined.
For example, if \( \cos x = 0.5 \), then \( \sec x = \frac{1}{0.5} = 2 \). However, the secant function isn't always defined. It's specifically undefined when \( \cos x = 0 \). Understanding this relationship is crucial, as it forms the basis for determining the domain of the secant function, which we will explore further.

Cosine Function

The cosine function, \( \cos x \), is a periodic function with a cycle of \( 2\pi \), meaning it repeats its values every \(2\pi\) radians. It is defined for all real numbers, and its values range from -1 to 1. Importantly, the cosine function reaches zero at specific points. These critical points occur at odd multiples of \( \frac{\pi}{2} \), such as \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \) or generally described by the expression \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer.
The significance of these points is that wherever \( \cos x \) equals zero, the reciprocal or secant function is undefined. These zeros of the cosine function create the critical boundaries for the domain of the secant function.

Domain of a Function

The domain of a function refers to all possible input values (or \( x \)-values) for which the function is defined. In simpler terms, it represents all the values you can plug into a function without causing an undefined expression. For the secant function, \( y = \sec x \), the domain consists of all real numbers except where the cosine function is zero.
Since \( \sec x = \frac{1}{\cos x} \), if \( \cos x = 0 \), then \( \sec x \) is undefined, as division by zero is not allowed. Therefore, for the secant function, the domain includes all real numbers except at \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer.

• The secant function has vertical asymptotes (lines where the function shoots up to infinity) at these excluded x-values.
• Every excluded point corresponds to a zero of the cosine function.

Understanding the domain helps graph the function correctly and anticipate its behavior across the x-axis.