Problem 51 - Precalculus: A Concise Course

Question

Use a graph to solve the equation on the interval \([-2 \pi, 2 \pi]\). $$ \cot x=-\frac{\sqrt{3}}{3} $$

Step-by-Step Solution

Answer

The solutions to the equation in the interval \([-2\pi, 2\pi]\) are \(-\frac{4\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3}\), and \frac{4\pi}{3}.

Step 1: Understanding the cotangent function

The cotangent function, denoted as \(\cot x\), is the reciprocal of the tangent function. It is periodic with a period of \(\pi\). In other words, \(\cot (x + n\pi) = \cot x\) for any integer \(n\). The function yields positive values in the fourth and the second quadrant while negative values in the first and the third quadrant.

Step 2: Recognizing the given value

It's been given that \(\cot x = -\frac{\sqrt{3}}{3}\). Recognizing this value is an important step. Now, \(-\frac{\sqrt{3}}{3}\) corresponds to the cotangent of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\), since in these angles cotangent function has a negative value.

Step 3: Constructing the solution set

Since cotangent is periodic with a period of \(\pi\), the solutions to the equation will be in the form: \(x = \frac{2\pi}{3} + n\pi\) and \(x = \frac{4\pi}{3} + m\pi\) where \(n\) and \(m\) are integers. The next step is to find the values of \(n\) and \(m\) such that the solutions fall into the given interval \([-2\pi, 2\pi]\). After substituting different values of \(n\) and \(m\), it's easy to see that the solutions in the given interval will be: \(-\frac{4\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3}\), and \(\frac{4\pi}{3}\).

Key Concepts

Cotangent Function

The cotangent function, represented as \( \cot x \), plays a crucial role in trigonometry. It is related to the tangent function as its reciprocal, meaning that \( \cot x = \frac{1}{\tan x} \). This makes it unique among trig functions, giving it distinctive properties.

Unlike sine and cosine, which have a period of \( 2\pi \), the cotangent function is periodic with a period of \( \pi \). This periodicity means that the pattern of the cotangent's graph repeats every \( \pi \) units. You can write this property as \( \cot(x + n\pi) = \cot x \) for any integer \( n \).

When examining its graph, the cotangent function takes on positive values in the fourth and second quadrants, while it is negative in the first and third quadrants, unlike tangent. This behavior is crucial when solving equations involving cotangent, as it helps predict the function's sign based on the angle's quadrant.

The understanding of these characteristics aids students in recognizing where possible solutions may lie on the unit circle, based on given values in an equation.

Periodic Functions

Periodic functions are a cornerstone of trigonometry and describe any function that repeats its values in regular intervals or cycles. Trigonometric functions like sine, cosine, and cotangent are classic examples due to their inherent cyclic nature.

The key attribute of periodic functions is their period, the length of one complete cycle of the function before it repeats. For cotangent, this period is \( \pi \). Thus, this property tells us that if we know the function's value at any point, it will return to this value after every interval of \( \pi \).

An essential application of understanding periodicity is to solve trigonometric equations over specified intervals, like \([-2\pi, 2\pi]\). It allows you to determine a finite set of solutions by considering how many cycles fit within the given range. When the periodic nature is leveraged efficiently, solving becomes a more straightforward process as you systematically identify the repeating solutions.

Graphical Solutions

Graphical solutions for trigonometric equations give a visual insight into where solutions lie, especially over specified intervals. It involves plotting the function and observing where it intersects with lines representing constant values.

For example, solving \( \cot x = -\frac{\sqrt{3}}{3} \) graphically would mean plotting the cotangent curve over the interval \([-2\pi, 2\pi]\). You then identify the horizontal line at \(-\frac{\sqrt{3}}{3}\) and pinpoint where this line intersects the cotangent curve within that interval.

This graphical approach simplifies the process of identifying solutions visually rather than purely algebraically, offering insight into the function's behavior and allowing for a clearer verification of potential algebraic missteps. Furthermore, visual aids help reinforce understanding of periodicity, quadrant-specific values, and the general shape of trigonometric functions.