Problem 106 - Calculus for Biology and Medicine

Question

Use the fact that \(\csc x=\frac{1}{\sin x}\) 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 domain of \(y = \csc x\) excludes integer multiples of \(\pi\) since \(\csc x = \frac{1}{\sin x}\) is undefined when \(\sin x = 0\).

Step 1: Understand the Function Relationship

The function \(y = \csc x\) is defined as the reciprocal of the sine function, \(\sin x\). So, we have \(\csc x = \frac{1}{\sin x}\). This means that the cosecant function is only defined when \(\sin x eq 0\).

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

The sine function equals zero at integer multiples of \(\pi\), i.e., \(\sin x = 0\) when \(x = n\pi\), where \(n\) is any integer. This is because the sine of any multiple of \(\pi\) (i.e., \(\sin(0)\), \(\sin(\pi)\), \(\sin(2\pi)\), etc.) is zero.

Step 3: Determine the Domain Restrictions

Given that \(\csc x\) is \(\frac{1}{\sin x}\), it is undefined wherever \(\sin x = 0\). Thus, the cosecant function cannot be defined at \(x = n\pi\), where \(n\) is an integer, as the function would lead to division by zero.

Step 4: State the Maximum Domain for \(y = \csc x\)

Considering that \(\csc x\) is undefined at \(n\pi\), the largest possible domain for \(y = \csc x\) consists of all real numbers except integer multiples of \(\pi\). This ensures the function is always defined and excludes the points where \(\sin x = 0\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental to trigonometry, helping us understand the relationship between the angles and sides of a triangle. These functions are crucial in various fields of science and engineering, particularly when analyzing waveforms and oscillatory phenomena.
One of the core trigonometric functions is the sine function, denoted as \(\sin x\), which measures the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The cosecant function, denoted as \(\csc x\), is the reciprocal of the sine function. Therefore, if \(\sin x\) is known, \(\csc x\) can be calculated as \(\csc x = \frac{1}{\sin x}\).

• When \(\sin x\) is positive, \(\csc x\) is also positive.
• When \(\sin x\) is negative, \(\csc x\) becomes negative
• As \(\sin x\) approaches zero, \(\csc x\) approaches infinity.

These basic relationships illustrate how tightly interconnected these functions are, and highlight the importance of the sine function in defining cosecant.

Domain of a Function

In mathematics, the domain of a function refers to the complete set of possible input values (often represented by \(x\)) for the function. It's important because it defines when and where the function exists and operates correctly.
For the cosecant function \(y = \csc x\), determining the domain requires focusing on when the function becomes undefined. Since \(y = \csc x\) is defined as \(\frac{1}{\sin x}\), it becomes problematic when \(\sin x = 0\), as division by zero is undefined in mathematics.

• \(\sin x = 0\) at integer multiples of \(\pi\) (i.e., \(x = n\pi\)).
• This means that at these points, \(\csc x\) does not exist.

Thus, the domain of the cosecant function includes all real numbers except these integer multiples of \(\pi\). This insight ensures the function remains consistently defined.

Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are relationships that express one trigonometric function in terms of the reciprocal of another. These identities are a powerful tool, simplifying complex trigonometric expressions and aiding in solving trigonometric equations.
The key reciprocal identities to remember are:

• \(\csc x = \frac{1}{\sin x}\)
• \(\sec x = \frac{1}{\cos x}\)
• \(\cot x = \frac{1}{\tan x}\)

Each reciprocal function complements its corresponding primary function. For instance, since \(\csc x\) is the reciprocal of \(\sin x\), it's undefined exactly where \(\sin x\) is zero—at every integer multiple of \(\pi\). These identities provide a framework for analyzing trigonometric behaviors, and understanding them is crucial for mastering trigonometry.