Problem 191 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \cot \frac{\pi}{4} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Definition of Cotangent

The cotangent of an angle is defined as the reciprocal of the tangent of that angle. So, \( \cot \theta = \frac{1}{\tan \theta} \).

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

The angle \( \frac{\pi}{4} \) is 45 degrees in the unit circle. At \( 45^\circ \), the tangent of the angle \( \tan \frac{\pi}{4} \) is 1, because \( \tan \theta \) is \( \frac{\text{opposite}}{\text{adjacent}} \), and both opposite and adjacent are equal in a 45-degree right triangle.

Step 3: Calculate the Cotangent

Using the definition from Step 1, \( \cot \frac{\pi}{4} = \frac{1}{\tan \frac{\pi}{4}} = \frac{1}{1} = 1 \).

Key Concepts

Cotangent Function

The cotangent function is a fundamental trigonometric function. It is defined as the reciprocal of the tangent function. This means that if you know the tangent of an angle, the cotangent is simply the inverse. Mathematically, it’s expressed as:

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

The cotangent function helps in determining the relationship between the sides of a right triangle, specifically between the adjacent side and the opposite side. When the tangent is easy to find, the cotangent can be quickly obtained using its definition.
The cotangent is often used in fields that require precise calculations, such as architecture and various engineering disciplines.

Tangent Function

The tangent function is another essential trigonometric function. It represents the ratio of the opposite side to the adjacent side in a right triangle. Formally, this is expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

For an angle \( \theta \) in the unit circle, the tangent function gives the slope of the terminal line with respect to the x-axis. When the angle \( \theta \) is \( \frac{\pi}{4} \) (or 45 degrees), the tangent equals 1. This is because the sides of the triangle are equal, turning the ratio \( \frac{1}{1} = 1 \).
The tangent is particularly useful in situations where you need to solve problems involving angles and distances, like in physics and robotics.

Unit Circle

The unit circle is a circle with a radius of one, centered on the origin of the coordinate plane. It’s a critical tool in trigonometry as it allows us to define trigonometric functions for all angles. For any angle \( \theta \), the coordinates of the corresponding point on the unit circle are \((\cos \theta, \sin \theta)\).
Using the unit circle, we can understand the cyclical nature of trigonometric functions. At \( 45^\circ \) or \( \frac{\pi}{4} \), both the sine and cosine are equal, which simplifies the calculation of tangent and cotangent.
The unit circle helps visualize how angles and trigonometric functions relate, making it easier to comprehend complex mathematical concepts.

Angle Conversion

Angles can be measured in different units, with degrees and radians being the most common. Conversion between these units is crucial for solving trigonometric problems correctly. To convert degrees into radians, you use the formula:

• \( \, \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).

Conversely, to change radians into degrees:

• \( \, \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \).

Understanding both units ensures flexibility in solving trigonometric problems and is particularly important in fields like physics and engineering, where radians are often preferred due to their natural properties with respect to calculus and periodicity.