Problem 18 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\cot \left(-\frac{\pi}{3}\right) \quad\) (b) \(\cot \frac{2 \pi}{3} \quad\) (c) \(\cot \frac{5 \pi}{3}\)

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{3}}{3}\), (b) \(-\frac{\sqrt{3}}{3}\), (c) \(-\frac{\sqrt{3}}{3}\)

Step 1: Trigonometric Function Basics

Recall that the cotangent function, \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]Utilize this definition to find the cotangent of any angle.

Step 1: Evaluate \( \cot\left(-\frac{\pi}{3}\right) \)

\(\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\). Therefore, \[\cot\left(-\frac{\pi}{3}\right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}\]which simplifies to \[\cot\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{3}\].

Step 2: Evaluate \( \cot \frac{2\pi}{3} \)

\(\cos\left(\frac{2\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}\) and \(\sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\). Therefore, \[\cot\left(\frac{2\pi}{3}\right) = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}\]which simplifies to \[\cot\left(\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{3}\].

Step 3: Evaluate \( \cot \frac{5\pi}{3} \)

\(\cos\left(\frac{5\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\). Therefore, \[\cot\left(\frac{5\pi}{3}\right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}\]which simplifies to \[\cot\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{3}\].

Key Concepts

Cotangent Function Overview

The cotangent of an angle, often symbolized as \( \cot(\theta) \), is a trigonometric function that represents the ratio of the cosine of an angle to the sine of that angle. Mathematically, it is expressed as:\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]This function is particularly useful for solving various trigonometric problems where you need to find the relationship between the sides of a right triangle. Since the cotangent is the reciprocal of the tangent function, it comes into play significantly in trigonometry together with the other primary trigonometric functions: sine, cosine, and tangent.When evaluating the cotangent for specific angles, it is helpful to recall or compute the values of sine and cosine for that angle. For angles in the unit circle, these values can often be retrieved from known values, such as \( \pi/3 \) or \( \pi/6 \). Calculating cotangent involves simple arithmetic of dividing the cosine value by the sine value for the angle in question.

Understanding Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for each angle value in their domain. These identities are useful tools in simplifying complex trigonometric expressions or solving trigonometric equations.For example, the identity for cotangent in terms of sine and cosine is:\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]Other fundamental identities include the Pythagorean identity, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \), which is pivotal in relating different trigonometric functions to each other. Additionally, the reciprocal identities clarify relationships like:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

Through these identities, you can transform and manipulate trigonometric expressions for various applications in algebra and calculus. They also play an essential role in evaluating trigonometric functions for angle values beyond the common angles utilized in earlier math courses.

Evaluating Angles in Trigonometric Functions

Evaluating trigonometric functions at specific angles often involves understanding the unit circle—a fundamental concept in trigonometry. The unit circle allows us to define trigonometric functions for all real numbers and helps in understanding their periodic nature.To evaluate \( \cot(-\pi/3) \), \( \cot(2\pi/3) \), and \( \cot(5\pi/3) \), it's useful to know certain things:

• The reference angle concept, which is the acute angle formed by the terminal side of a given angle and the x-axis.
• Symmetry in the unit circle means angles in different quadrants with the same reference angle have sine and cosine values differing only by sign.

For instance, \( \cot(-\pi/3) \) leverages the reference angle \( \pi/3 \), with symmetry to find \( \cos(-\pi/3) = \cos(\pi/3) = 1/2 \) and \( \sin(-\pi/3) = -\sin(\pi/3) = -\sqrt{3}/2 \), leading to \( \cot(-\pi/3) = -\sqrt{3}/3 \).Understanding how to find an angle's sine and cosine on the unit circle, and how their ratios reveal the cotangent, simplifies the process of evaluating these trigonometric expressions. Each angle follows rules dictated by its position (or quadrant) on the unit circle.