Problem 22 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc 30^{\circ}$$

Step-by-Step Solution

Answer

\(\csc 30^{\circ} = 2\); it is a rational number.

Step 1: Understand the Cosecant Function

The cosecant function, represented as \(\csc\), is the reciprocal of the sine function. This means \(\csc \theta = \frac{1}{\sin \theta}\). In this problem, \(\theta = 30^{\circ}\).

Step 2: Find \(\sin 30^{\circ}\)

Recall that \(\sin 30^{\circ}\) is a well-known angle in trigonometry with a value. Specifically, \(\sin 30^{\circ} = \frac{1}{2}\).

Step 3: Calculate \(\csc 30^{\circ}\) Using the Reciprocal

Using the definition of the cosecant function as the reciprocal of sine, we have:\[ \csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{\frac{1}{2}} = 2. \]

Step 4: Determine if the Value is Rational or Irrational

The number 2 is a rational number because it can be written as a simple fraction (\(\frac{2}{1}\)). Thus, no further decimal approximation is needed.

Key Concepts

Cosecant Function

The cosecant function is an essential part of trigonometry. It is often denoted as \( \csc \theta \) and is defined as the reciprocal of the sine function. This means:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Understanding this reciprocal relationship is crucial for solving trigonometric problems.
In elementary trigonometry, sine, cosine, and tangent are the primary functions. Cosecant belongs to a category called reciprocal trigonometric functions, which we'll discuss next. By knowing the sine of an angle, you can immediately find the cosecant by taking the reciprocal. For example, if \( \theta = 30^{\circ} \), and \( \sin 30^{\circ} = \frac{1}{2} \), then:

• \( \csc 30^{\circ} = \frac{1}{\frac{1}{2}} = 2 \)

This makes the cosecant calculation very straightforward.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic sine, cosine, and tangent functions. Not only do they help in reversing these functions, but they are also useful in various mathematical and engineering contexts. The three primary reciprocal trigonometric functions are:

• Cosecant (\( \csc \theta = \frac{1}{\sin \theta} \))
• Secant (\( \sec \theta = \frac{1}{\cos \theta} \))
• Cotangent (\( \cot \theta = \frac{1}{\tan \theta} \))

Reciprocal functions can also be represented on the unit circle, helping to visualize their values across different angles. Each of these functions provides insights into the behavior of the original trigonometric functions in inverse situations.
When solving problems like \( \csc 30^{\circ} \), recognizing these reciprocal relationships simplifies the process by allowing us to switch from one function to another using a simple division. This mechanism highlights the interconnectedness of trigonometric functions.
It’s important to remember that reciprocal functions often deal with undefined situations, such as when the original function approaches zero. This allows students to understand tackling undefined expressions, an essential skill in mathematics.

Exact Values in Trigonometry

Exact values in trigonometry refer to the known values of trigonometric functions at specific common angles like \( 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \( 90^{\circ} \). These values are well-documented, and having them memorized can be invaluable for quick calculations.
For example, some common exact trigonometric values include:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \tan 45^{\circ} = 1 \)
• \( \csc 60^{\circ} = \frac{2}{\sqrt{3}} \)

These precise values arise from geometric interpretations or the unit circle. They make solving trigonometric equations much more manageable, especially when determining rationality.
Consider \( \csc 30^{\circ} \), where \( \sin 30^{\circ} = \frac{1}{2} \). This allows the exact value for \( \csc 30^{\circ} \) to be calculated precisely as 2, a rational number because it can be expressed as a fraction (\( \frac{2}{1} \)).
Therefore, using exact values not only simplifies calculations but also helps verify results without relying solely on a calculator.