Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot x=-1$$

Step-by-Step Solution

Answer

The solutions of the equation \(\cot x = -1\) in the interval \([0, 2\pi)\) are \(x = 3\pi/4, 7\pi/4\).

Step 1: Identify Values where Cotangent is -1

The cotangent of an angle is -1 at \(-3\pi/4\), but since the interval is \([0,2\pi)\), this solution is not within range. The cotangent of an angle is also -1 at \(3\pi/4\). So \(3\pi/4\) is one of the solutions.

Step 2: Find Additional Solutions

The cotangent function has a period of \(\pi\), so it will repeat every \(\pi\) units. Adding \(\pi\) to the found solution of \(3\pi/4\) gives another solution within the defined interval: \(3\pi/4 + \pi = 7\pi/4\).

Step 3: Confirm Solutions are in Interval

The solutions \(3\pi/4\) and \(7\pi/4\) are both in the interval \([0, 2\pi)\) and are therefore the solutions of the equation in this interval.

Key Concepts

Trigonometric Equations

A trigonometric equation is an equation that involves trigonometric functions of an unknown angle. To solve such equations, we search for all the angles that make the equation true. When trying to find all solutions to a trigonometric equation, it is important to consider the domain or interval where the solution must lie.

For example, in the case of the equation \(\text{cot } x = -1\), the domain given is \([0, 2\pi)\). This is crucial when solving for \(x\) because trigonometric functions, including cotangent, can have infinite solutions over the real numbers. But considering the specific interval helps narrow them down to those that apply to the given situation. A systematic approach to solve these trigonometric equations includes finding a particular solution and then using the properties of trigonometric functions to find all such solutions within the specified interval.

Cotangent Function

The cotangent function, abbreviated as \(\text{cot}\), relates to the angle of a right triangle in relation to the ratio of the adjacent side over the opposite side. Its graph is not as common as sine and cosine, which means it can be slightly trickier for students to grasp. However, knowing that cotangent is the reciprocal of the tangent (which itself is the ratio of opposite over adjacent), can be quite helpful.

To understand the cotangent's behavior, remember it is undefined whenever the tangent is zero (since you cannot divide by zero), and it has a value of zero at odd multiples of \(\frac{\pi}{2}\). It's also important to recognize that cotangent is positive in the first and third quadrants where both tangent's numerator (the adjacent side) and denominator (the opposite side) have the same sign and negative in the second and fourth quadrants.

Trigonometry Intervals

When working with trigonometric functions, the concept of intervals is vital. An interval specifies the range of angle values we are considering. For instance, the interval \([0, 2\pi)\) includes all possible angles from \(0\) radians up to, but not including, \(2\pi\) radians, which corresponds to a full rotation around the unit circle.

In the context of the problem presented, we look for cotangent values in this interval. Since trigonometric functions repeat their values periodically, intervals help us limit the solutions to a manageable set, rather than dealing with infinitely many solutions that extend in both the positive and negative directions along the real number line.

Periodicity of Trigonometric Functions

The periodicity of trigonometric functions refers to the fact that these functions repeat their values at regular intervals. Each trigonometric function has a period, which is the length of the smallest interval after which the function starts repeating its pattern.

The cotangent function has a period of \(\pi\) radians. This means that once you find a particular solution to a cotangent equation, you can find additional solutions by adding or subtracting multiples of \(\pi\). In practice, when solving for \(x\) in an equation like \(\text{cot } x = -1\), after finding the initial solution within the given interval, the periodic nature of the cotangent function allows us to add \(\pi\) to the known solution to find subsequent solutions. This process continues until the added solutions are outside of the specified interval, ensuring that we have all unique solutions within the given domain.