Problem 29 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\cot \left(-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The exact value is -1.

Step 1: Identify the Right Quadrant

The angle \(-\frac{\pi}{4}\) is measured in the clockwise direction from the positive x-axis because it is negative. This places it in the fourth quadrant.

Step 2: Determine the Reference Angle

To find the reference angle for \(-\frac{\pi}{4}\), consider the angle's absolute value, which is \(\frac{\pi}{4}\). This reference angle lies on the 45-degree diagonal in the coordinate plane.

Step 3: Use Cotangent Definition

The cotangent of an angle \(\theta\) is defined as \(\cot(\theta) = \frac{1}{\tan(\theta)}\). For \(\theta = \frac{\pi}{4}\), \(\tan\left(\frac{\pi}{4}\right) = 1\), so \(\cot\left(\frac{\pi}{4}\right) = 1\).

Step 4: Adjust Based on Quadrant

In the fourth quadrant, the tangent function is negative, making \(\tan(-\frac{\pi}{4}) = -1\). Hence, \(\cot(-\frac{\pi}{4}) = \frac{1}{-1} = -1\).

Key Concepts

Cotangent

The cotangent is one of the six fundamental trigonometric functions, closely related to the tangent function. It represents the reciprocal of the tangent. The formula for cotangent is given as follows:

• For any angle \( \theta \), the cotangent is \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

In simpler terms, if you know the tangent value of an angle, you simply take its reciprocal to find the cotangent.
For angles like \( \frac{\pi}{4} \), where the tangent is \( 1 \), the cotangent becomes \( 1 \) since the reciprocal of \( 1 \) is also \( 1 \).
Understanding cotangent is crucial when solving problems involving trigonometric functions, as it helps in determining the relationship between different angles, especially in coordinate geometry.

Reference Angle

The reference angle is the smallest angle that a given angle shares with the nearest x-axis. It is always positive and measures the closeness of an angle to the x-axis.

• For any angle \( \theta \), the reference angle is the absolute measure of \( \theta \) between \( 0 \) and \( \frac{\pi}{2} \).
• In practical terms, the reference angle helps us to simplify calculations by giving us an equivalent angle in the first quadrant.

For example, the angle \(-\frac{\pi}{4}\) has a reference angle of \(\frac{\pi}{4}\), as we consider only the magnitude for determining the reference angle.
Using reference angles is a handy method to determine the value of trigonometric functions for different angles based on their position in any of the quadrants.

Quadrants

In the coordinate plane, the plane is divided into four quadrants, each representing different ranges of angles:

• The first quadrant, where both sine and cosine values are positive.
• The second quadrant, where sine is positive, but cosine is negative.
• The third quadrant, where both sine and cosine values are negative.
• The fourth quadrant, where cosine is positive, but sine is negative.

The position of an angle is determined by its measure positive or negative, hence identifying the correct quadrant.
For the angle \(-\frac{\pi}{4}\), since it's measured in the clockwise direction, it resides in the fourth quadrant. Here, the tangent function becomes negative, which immediately influences the cotangent, resulting in a negative value for \( \cot(-\frac{\pi}{4}) \), which is \(-1\).
Understanding quadrants is essential in trigonometry as it helps in predicting the signs of various trigonometric functions.