Problem 34 - 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. $$\cot \frac{\pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \( \cot \frac{\pi}{4} \) is 1, which is rational.

Step 1: Understanding Cotangent Function

The cotangent function is defined as the reciprocal of the tangent function. So, \( \cot \theta = \frac{1}{\tan \theta} \). Here, we need to find \( \cot \frac{\pi}{4} \).

Step 2: Calculate Tangent of \( \frac{\pi}{4} \)

Evaluate \( \tan \frac{\pi}{4} \). The tangent of \( \frac{\pi}{4} \) is 1 because, in a unit circle, \( \tan \theta \) is the ratio \( \frac{\text{opposite}}{\text{adjacent}} \) and for \( \theta = \frac{\pi}{4} \), both sides are equal resulting in \( \frac{1}{1} = 1 \).

Step 3: Compute Cotangent from Tangent

Using \( \cot \theta = \frac{1}{\tan \theta} \), substitute \( \tan \frac{\pi}{4} = 1 \). Hence, \( \cot \frac{\pi}{4} = \frac{1}{1} = 1 \).

Step 4: Exact Value

The exact value of \( \cot \frac{\pi}{4} \) is 1. Since 1 is a rational number, there is no need for decimal approximation using a calculator.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is a fundamental trigonometric function. It represents the reciprocal of the tangent function. This means, \( \cot \theta = \frac{1}{\tan \theta} \). For angles where the tangent function is known, the cotangent can be easily calculated by simply taking the reciprocal of the tangent value.
For example:

• If \( \tan \theta = 2 \), then \( \cot \theta = \frac{1}{2} \).
• If \( \tan \theta = \frac{1}{3} \), then \( \cot \theta = 3 \).

Understanding this relationship helps in various trigonometric calculations especially when dealing with angle measurements in different units like radians and degrees. The cotangent function is used in solving many trigonometric identities and equations.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the primary trigonometric functions. It measures the ratio of the opposite side to the adjacent side in a right-angled triangle. In the realm of the unit circle, \( \tan \theta \) is defined as \( \frac{\sin \theta}{\cos \theta} \).
Let’s consider an important angle:

• When \( \theta = \frac{\pi}{4} \), the sine and cosine values are equal, leading to \( \tan \frac{\pi}{4} = 1 \).

This property is especially helpful when working with special angle values. It's crucial in trigonometry as it assists in determining angles, solving triangle problems, and exploring periodic properties of trigonometric functions.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. This circle is invaluable in trigonometry as all the trigonometric functions can be derived from it. Points on the unit circle are of the form \((\cos \theta , \sin \theta)\).
Here's why it’s important:

• The angle \( \theta \) in the circle corresponds to the arc length on this circle.
• For any angle, \( \tan \theta \) is represented as the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

Understanding the unit circle allows one to calculate trigonometric values even without a calculator, especially for standard angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and so on.

Exact Value Calculation

Calculating the exact value of trigonometric functions involves using known values that do not require approximation. For certain angles, like \( \frac{\pi}{6} \), \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and others, the trigonometric values are well-documented and require no approximation.
Consider:\[ \cot \frac{\pi}{4} = 1 \]
This is derived directly from the reciprocal of \( \tan \frac{\pi}{4} \), which equals to 1. Since 1 is a rational number, it is already known exactly, making decimal approximation unnecessary. These exact values are integral when solving trigonometric equations or identities without error stemming from approximations.