Problem 102 - Calculus for Biology and Medicine

Question

Use the fact that \(\cot x=\frac{1}{\tan x}\) to explain why the maximum domain of \(y=\csc x\) consists of all real numbers except integer multiples of \(\pi\).

Step-by-Step Solution

Answer

The maximum domain of \(y=\csc x\) includes all real numbers except integer multiples of \(\pi\) since \(\csc x\) is undefined where \(\sin x = 0\), which occurs at these multiples.

Step 1: Understand the Function

The function we are analyzing is the cosecant function, denoted as \(y = \csc x\). Recall that \(\csc x = \frac{1}{\sin x}\). This indicates that the function is only defined where \(\sin x eq 0\).

Step 2: Identify When \(\sin x = 0\)

The sine function, \(\sin x\), equals zero at integer multiples of \(\pi\). Mathematically, this means \(x = n\pi\) where \(n\) is an integer. These are the locations on the x-axis where the sine wave crosses the axis.

Step 3: Eliminate Undesirable Points from Domain

Since \(\csc x = \frac{1}{\sin x}\) is undefined wherever \(\sin x = 0\), and we've identified these points as integers multiples of \(\pi\), we exclude these from the domain of \(\csc x\). Therefore, any real number that isn't a multiple of \(\pi\) is part of the domain.

Key Concepts

Understanding the Cosecant Function

The cosecant function, denoted as \( y = \csc x \), is fundamentally tied to the sine function. In fact, it is the reciprocal of the sine function, which means that \( \csc x = \frac{1}{\sin x} \). This relationship sets the stage for analyzing when and where this function is defined.

• The cosecant function exists wherever the sine function does not equal zero. This is because having a zero in the denominator makes the expression undefined.
• As a result, to find the domain of the cosecant function, we need to look closely at the points where the sine function equals zero.

Understanding this reciprocal nature is crucial, as it reveals when the cosecant function does not exist, directly impacting its domain.

Key Aspects of the Sine Function

To unlock the behavior of the cosecant function, it's important to grasp how the sine function operates. The sine function, \( \sin x \), is a periodic function with several key characteristics:

• Its values range from \(-1\) to \(1\).
• It has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.
• The sine function equals zero at specific points: specifically, integer multiples of \(\pi\), such as \(0, \pi, 2\pi, -\pi\), etc.

When Does the Sine Equal Zero?
The sine function equals zero at points \( x = n\pi \), where \( n \) is any integer. At these points, the sine wave crosses the x-axis, making the cosecant function undefined because it would involve division by zero. Therefore, these specific points are excluded from the domain of the cosecant function.

Exploring the Domain of a Function

The domain of a function refers to all the input values (or x-values) for which the function is defined.

• For most functions, you look to avoid values that make the function undefined, such as division by zero or the square root of a negative number.
• For \( y = \csc x \), since \( \csc x = \frac{1}{\sin x} \), the function becomes undefined wherever \( \sin x = 0 \).

Setting the Domain of the Cosecant Function
To establish the domain of \( y = \csc x \), we exclude any x-value that can be expressed as \( n\pi \), where \( n \) is an integer. This is because these are the exact points where \( \sin x = 0 \), rendering the cosecant undefined.

• Thus, the domain of \( y = \csc x \) consists of all real numbers except integer multiples of \( \pi \).

Understanding the domain helps clarify where and when you can use the function without running into undefined regions.