Problem 374 - Precalculus

Question

Find the exact value of \(\cot \frac{\pi}{4}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Cotangent Function

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

Step 2: Recognizing Key Angles

The angle \( \frac{\pi}{4} \) radians corresponds to 45 degrees. It is a key angle in trigonometry with known values for sine and cosine. For \( \frac{\pi}{4} \), both \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \) are equal to \( \frac{\sqrt{2}}{2} \).

Step 3: Calculating Tangent for \( \frac{\pi}{4} \)

The tangent function is defined as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, \( \tan \frac{\pi}{4} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \).

Step 4: Finding Cotangent from Tangent

Since \( \cot \theta = \frac{1}{\tan \theta} \), we can find \( \cot \frac{\pi}{4} \) using \( \tan \frac{\pi}{4} \). Thus, \( \cot \frac{\pi}{4} = \frac{1}{1} = 1 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental elements in mathematics, particularly in geometry and calculus. They help describe the relationships between the angles and sides of triangles.
These functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), tangent (\( \tan \theta \)), cotangent (\( \cot \theta \)), secant (\( \sec \theta \)), and cosecant (\( \csc \theta \)). Each of these functions provides a unique perspective on the triangle's properties.
Specifically, the cotangent function, which is the focus here, can be defined as the reciprocal of the tangent function. That means:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Understanding these basic relationships is crucial. They set the stage for solving more complex problems in trigonometry.

Key Angles in Trigonometry

Key angles in trigonometry are special angles with values that are easy to remember and often used as benchmarks in calculations. These angles usually include 0°, 30°, 45°, 60°, and 90°, and their respective radian measures, such as \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
These angles are significant because:

• They have simple sine and cosine values that often repeat.
• They are essential for defining the unit circle, which helps visualize trigonometric functions.

For example, the angle \( \frac{\pi}{4} \), also known as 45°, has both sine and cosine values of \( \frac{\sqrt{2}}{2} \). This symmetry results in a tangent of 1 because \( \tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = 1 \). Recognizing these angles and their properties can simplify the understanding of more intricate concepts.

Reciprocal Identities

Reciprocal identities are relationships that relate certain trigonometric functions through inverse operations. These identities play a crucial role in simplifying trigonometric expressions and solving equations.
The reciprocal identities involve:

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

In the case of cotangent, its relationship as the reciprocal of tangent means that if we know \( \tan \theta \), finding \( \cot \theta \) is straightforward as shown: \( \cot \theta = \frac{1}{\tan \theta} \).
This relationship is handy when working with angles like \( \frac{\pi}{4} \), where \( \tan \frac{\pi}{4} = 1 \), so \( \cot \frac{\pi}{4} = \frac{1}{1} = 1 \). Mastering these reciprocal identities can dramatically enhance problem-solving skills in trigonometry.