Problem 31 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Cotangent Function

The cotangent of an angle \( \theta \) is defined as \( \cot \theta = \frac{1}{\tan \theta} \). It is the reciprocal of the tangent function. \(-\frac{\pi}{4}\) is the angle whose cotangent needs to be found.

Step 2: Identify the Tangent of the Angle

First, find \( \tan \left(-\frac{\pi}{4}\right) \). Since tangent is an odd function, \( \tan(-\theta) = -\tan(\theta) \). \( \tan \left(\frac{\pi}{4}\right) = 1 \), so \( \tan \left(-\frac{\pi}{4}\right) = -1 \).

Step 3: Calculate the Cotangent Using the Tangent

Now, using \( \cot \theta = \frac{1}{\tan \theta} \), we can find the cotangent of the angle: \( \cot \left(-\frac{\pi}{4}\right) = \frac{1}{-1} = -1 \).

Key Concepts

Cotangent

The cotangent function is a fundamental concept in trigonometry. Essentially, the cotangent is the reciprocal of the tangent function. This means we can express it as:

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

The cotangent function is particularly useful when dealing with right triangles or circles to find ratios involving adjacent and opposite sides.
A quick way to remember this is: cotangent is equal to adjacent over opposite. However, when using the unit circle, it's handy to think about it as the ratio based on \(x\) and \(y\) coordinates for angles.
The cotangent function has some unique properties:

• Values are undefined where the tangent is zero.
• It is periodic with a period of \(\pi\).
• It is also asymmetric, making studying its behavior interesting as the angle changes.

By understanding its definition and properties, we can calculate values easily like in the exercise above.

Tangent Function

The tangent function is one of the six primary trigonometric functions and can be defined in the simplest terms as the ratio of sine and cosine:

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

Its graph is characterized by periodic vertical asymptotes and repeating patterns, making it a periodic function with period \(\pi\).
When evaluating the tangent of an angle like \(-\frac{\pi}{4}\), it helps to note that:

• For angles in radians, \(\tan \left(\frac{\pi}{4}\right) = 1\)

Furthermore, recognizing that tangent is an odd function (

• \(\tan(-\theta) = -\tan(\theta)\)

), is crucial because it helps determine the tangent of its negative angle simply by changing the sign of the result.
This property of odd functions greatly simplifies the computation process in our exercise, allowing us to find \(\tan\left(-\frac{\pi}{4}\right) = -1\).

Odd Function

Odd functions are a unique set of mathematical functions with characteristic symmetry about the origin of the graph. When graphed, they exhibit a reflectional symmetry if one reflects them over both the x-axis and y-axis.
Mathematically, a function is considered odd if:

• \(f(-x) = -f(x)\) for all \(x\)

In trigonometry, several functions, including sine and tangent, are odd. This property simplifies calculations greatly.
For our trigonometric identities:

• Sine and tangent functions preserve the odd function property.
• For example, \(\tan(-x) = -\tan(x)\), which was used in the exercise above.

Thus, understanding the odd function concept can make solving angles and transformations more intuitive, especially when paired with periodic functions like tangent. Knowing this can help anticipate the behavior of the function under negative angles.