Problem 31 - 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. $$\csc \frac{\pi}{6}$$

Step-by-Step Solution

Answer

The exact value is 2; it's a rational number.

Step 1: Understand the Problem

We need to find the cosecant of \( \frac{\pi}{6} \). The cosecant function is the reciprocal of the sine function. So, \( \csc \theta = \frac{1}{\sin \theta} \). Our task is to calculate \( \csc \frac{\pi}{6} \).

Step 2: Recall Sine Value for \( \frac{\pi}{6} \)

The sine of \( \frac{\pi}{6} \) is a well-known angle in trigonometry. Specifically, \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 3: Calculate Cosecant

Since \( \csc \theta = \frac{1}{\sin \theta} \), we can find \( \csc \frac{\pi}{6} \) by taking the reciprocal of \( \sin \frac{\pi}{6} \). Therefore, \( \csc \frac{\pi}{6} = \frac{1}{\frac{1}{2}} = 2 \).

Step 4: Confirm the Value is Rational

Since \( \csc \frac{\pi}{6} = 2 \), this value is rational as it is a whole number. There's no need to use a calculator for decimal approximation in this case.

Key Concepts

Cosecant Function

In trigonometry, the cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. This means that for any angle \( \theta \), \( \csc \theta = \frac{1}{\sin \theta} \). It is an important function used to understand the properties of triangles and circles.
A few key facts about the cosecant function include:

• It is only defined for angles where the sine is not zero because division by zero is undefined.
• The cosecant function tends to infinity as the angle approaches values where the sine is zero.
• To calculate the cosecant, simply find the sine of the angle and take its reciprocal.

Understanding the relationship between sine and cosecant is crucial in solving various trigonometric problems, especially when dealing with trigonometric identities and angle calculations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are a set of functions derived from the primary trigonometric functions by taking their reciprocals. There are three primary reciprocal trigonometric functions:

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

These functions are particularly useful in certain equations and can offer simpler forms in specific scenarios. They also extend trigonometric function values to a broader range of real numbers by encouraging the use of angles that might initially seem problematic, like when the sine, cosine, or tangent is zero for the primary functions.
Importantly, understanding reciprocal trigonometric functions provides a deeper comprehension of trigonometry as they often appear in calculus, complex numbers, and other mathematical applications.

Pi Radians

The term "pi radians" refers to a measure of angles where the entire circle is considered to have an angle of \( 2 \pi \) radians. Understanding radians is essential in trigonometry as they provide a more natural measurement of angles compared to degrees.
For instance, \( \pi \) radians equal 180 degrees, so when you see \( \frac{\pi}{6} \), it can be converted to 30 degrees by multiplying by \( \frac{180}{\pi} \). This is a common practice because many calculators and mathematical formulas are based on radians rather than degrees.

• Radians help simplify formulas in calculus and other applied mathematics areas.
• Many trigonometric identities and formulas rely on radian measurements.
• Understanding radians enhances your mathematical fluency and enables seamless transitions between various branches of mathematics.

Overall, radians, and especially pi radians, offer a standardized method for expressing angles, making them invaluable in advanced mathematical problem-solving.