Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cot \left(-\frac{4 \pi}{7}\right)$$

Step-by-Step Solution

Answer

\(-\cot\left(\frac{4\pi}{7}\right)\)

Step 1: Understanding Even-Odd identities

Trigonometric identities classify functions as even or odd. For example, cotangent function satisfies the property of an odd function, which means that for any angle \( heta \), \( \cot(-\theta) = -\cot(\theta) \).

Step 2: Apply Odd Identity to Cotangent

Given \( \cot\left(-\frac{4\pi}{7}\right) \), apply the odd function property: \( \cot\left(-\frac{4\pi}{7}\right) = -\cot\left(\frac{4\pi}{7}\right) \). Thus, the expression is simplified to \( -\cot\left(\frac{4\pi}{7}\right) \).

Step 3: Express as a Function of a Positive Angle

The expression \( -\cot\left(\frac{4\pi}{7}\right) \) is the positive angle form of the original expression, as \( \frac{4\pi}{7} \) is already positive.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially when dealing with angles and periodic phenomena. They originate from the study of triangles, specifically right-angled triangles. Here are some of the basic trigonometric functions:

• Sine (\( \sin \theta \)): Opposite side divided by the hypotenuse.


• Cosine (\( \cos \theta \)): Adjacent side divided by the hypotenuse.


• Tangent (\( \tan \theta \)): Opposite side divided by the adjacent side.

Each of these functions can take positive or negative values depending on the angle and its position in the coordinate system.
In addition, there are reciprocal functions like cosecant, secant, and cotangent that provide additional ways to relate angles and sides of triangles.
Trigonometric functions also have periodic properties, meaning they can repeat values at regular intervals, which makes them incredibly useful for modeling cycles and waves.

Odd Functions

In mathematical terms, a function is considered 'odd' if it satisfies the condition \( f(-x) = -f(x) \).
This means that when you substitute any number with its negative equivalent, the function's output will also change signs.
This symmetry can be visualized when the graph of an odd function is rotated 180 degrees about the origin, it remains unchanged.
Here are some characteristics and examples:

• Odd functions appear symmetric across the origin.


• Common odd functions include: Sine (\( \sin(-x) = -\sin(x) \)), Tangent (\( \tan(-x) = -\tan(x) \)), and Cotangent (\( \cot(-x) = -\cot(x) \)).

Odd functions play a crucial role in solving trigonometric identities because they simplify the expressions and verify their correctness.
The cotangent function's odd nature was key to transforming \( \cot\left(-\frac{4\pi}{7}\right) \) into its equivalent at a positive angle.

Cotangent Function

The cotangent function, often represented as \( \cot \theta \), is the reciprocal of the tangent function.
Mathematically, it is written as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
This function is useful in various applications, especially when dealing with angles in different quadrants.

• Cotangent is an odd function, meaning \( \cot(-\theta) = -\cot(\theta) \).


• It is undefined at angles where sine is zero (multiples of \( \pi \)).


• Cotangent has a period of \( \pi \) which means it repeats its values every \( \pi \) radians.

Understanding the properties of cotangent helps in transforming expressions like \( \cot\left(-\frac{4\pi}{7}\right) \) into simpler forms.
By recognizing its odd nature, you can express it as \( -\cot\left(\frac{4\pi}{7}\right) \), completing the transformation to a positive angle.