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: Identify the Even-Odd Identity

For cotangent, we use the identity \( \cot(-x) = -\cot(x) \). This tells us that the cotangent function is an odd function, which means that \( \cot\left(-\frac{4\pi}{7}\right) = -\cot\left(\frac{4\pi}{7}\right) \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

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

Key Concepts

Even-Odd Identities

In trigonometry, functions can be classified as either even or odd, based on their symmetrical properties. Even functions have symmetry with respect to the y-axis. Conversely, odd functions have symmetry with respect to the origin. This means that if you flip the graph of an odd function over both the x-axis and the y-axis, it stays the same.

Understanding even-odd identities helps to simplify expressions and solve trigonometric equations. The identity for the sine function, for example, is

• even: none
• odd: \( \sin(-x) = -\sin(x) \)

While for the cosine function, it's:

• even: \( \cos(-x) = \cos(x) \)
• odd: none

Trigonometric functions that exhibit odd symmetry, like the sine and tangent functions, follow the formula \( f(-x) = -f(x) \). Recognizing whether a function is even or odd simplifies working with negative angles, as seen in the problem above with the cotangent function, which is an odd function.

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions. It can be defined as the reciprocal of the tangent function, which is given by:\[ \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \]This means that the cotangent of an angle is the ratio of the adjacent side over the opposite side in a right-angled triangle. The cotangent function is less commonly used than sine or cosine, but it is particularly useful in certain mathematical and engineering applications.

One crucial property of cotangent is that it is an odd function. This is encapsulated by the identity \( \cot(-x) = -\cot(x) \). In practical terms, this identity allows for the easy conversion of cotangent values of negative angles into their positive counterparts. Understanding this property aids in solving trigonometric equations involving cotangent, especially when working with transformations or phase shifts.

Trigonometric Functions

Trigonometric functions are functions of an angle commonly used in mathematics, physics, and engineering. These functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are:

• Sine (\( \sin \))
• Cosine (\( \cos \))
• Tangent (\( \tan \))
• Cotangent (\( \cot \))
• Secant (\( \sec \))
• Cosecant (\( \csc \))

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. This periodicity is a powerful tool for analyzing wave patterns and circular motion.

Each function has unique properties and identities. For instance, sine and cosine are periodic with a period of \( 2\pi \), while tangent and cotangent have a period of \( \pi \). Real-world applications of trigonometric functions include signal processing, where they model sound and light waves, and in navigation, where they help in position determination and waypoint calculations. By understanding these functions and their identities, we simplify complex mathematical tasks and unlock solutions to problems in various domains.