Problem 11 - 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 $$\tan \left(-\frac{\pi}{7}\right)$$

Step-by-Step Solution

Answer

\(-\tan\left(\frac{\pi}{7}\right)\)

Step 1: Identify the Even-Odd Identity

The tangent function is an odd function. This means that for any angle \( \theta \), \( \tan(-\theta) = -\tan(\theta) \). We will apply this property to simplify the given expression.

Step 2: Apply the Odd Property

Given the expression \( \tan\left(-\frac{\pi}{7}\right) \), apply the odd function identity: \( \tan\left(-\frac{\pi}{7}\right) = -\tan\left(\frac{\pi}{7}\right) \).

Key Concepts

Tangent Function

The tangent function is a fundamental part of trigonometry. It relates the angles in a right-angled triangle to the ratios of two of its sides. Specifically, the tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side of the triangle:
- \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
This function helps solve various geometry problems and describes wave patterns and oscillations in physics. Tangent, along with sine and cosine, creates the primary foundation for trigonometric functions, extending several applications beyond triangles into circular functions. Additionally, tangent has a critical role in identities and properties essential for simplifying expressions.
Beyond basic geometric interpretations, the tangent function is periodic and has asymptotes where the function is undefined. These aspects are vital when dealing with problems involving larger angles or their identities.

Odd Function Property

An important concept to understand in trigonometry is the classification of functions into even and odd. A function \( f(x) \) is an odd function if it satisfies the property:

• \( f(-x) = -f(x) \)

For the tangent function, this property confirms that it is indeed an odd function. When examining \( \tan(\theta) \), applying this property, we have:
- \( \tan(-\theta) = -\tan(\theta) \)
This identity becomes incredibly useful when dealing with expressions involving negative angles. In such cases, it allows for the transformation of the problem into one involving a positive angle, thereby simplifying the computation or analysis necessary. This is precisely what happens in transforming \( \tan\left(-\frac{\pi}{7}\right) \) to \(-\tan\left(\frac{\pi}{7}\right) \), a much easier expression to handle.
Understanding the odd function property and its application is crucial for simplifying trigonometric problems, as it directly influences how equations and expressions are solved in a more convenient manner.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities and properties of trigonometric functions. By applying these known identities, complex expressions can be transformed into simpler forms. This not only makes computation easier but also aids in discovering deeper relationships and patterns within the math.For example, with the exercise expression \( \tan\left(-\frac{\pi}{7}\right) \), the odd function property allows transformation into \(-\tan\left(\frac{\pi}{7}\right) \). These transformations help to work with non-standard angles, often making them more manageable. Using trig identities like Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), angle addition formulas, and others also support these simplification processes.
When solving equations, especially in physics or engineering, simplifying trigonometric expressions is an indispensable technique. It helps extract meaningful results or solutions and reduces computational complexity, reflecting the importance of mastering these strategies within broader scientific or mathematical problems.