Problem 77 - Precalculus

Question

Solve the equation \(\tan (2 x-\pi)=0\) for \(x\) in the interval \([-\pi, \pi]\) by graphing.

Step-by-Step Solution

Answer

The solutions are \( x = -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi \).

Step 1: Identify the Tangent Function

The given equation is \( \tan (2x - \pi) = 0 \). With the nature of the \( \tan \theta \) function, it equals zero whenever the angle \( \theta = n\pi \), where \( n \) is an integer.

Step 2: Set Up the Equation for Zeros

Given \( \tan (\theta) = 0 \) when \( \theta = n\pi \), we set \( 2x - \pi = n\pi \). Thus, \( 2x = n\pi + \pi \) or \( 2x = (n+1)\pi \).

Step 3: Solve for x

Now, divide everything by 2 to solve for \( x \): \[ x = \frac{(n+1)\pi}{2} \]

Step 4: Determine Valid n for Interval

We need \( x \) values to be within the interval \([ -\pi, \pi] \). Substituting back we have: \[ -\pi \leq \frac{(n+1)\pi}{2} \leq \pi \] Multiply through by 2: \[ -2\pi \leq (n+1)\pi \leq 2\pi \] Divide by \( \pi \): \[ -2 \leq n+1 \leq 2 \]Rewriting the inequality gives \[ -3 \leq n \leq 1 \] Thus, \( n \) can take values \(-3, -2, -1, 0, 1\).

Step 5: Calculate Corresponding x Values

With values of \( n \) as \(-3, -2, -1, 0, 1\), calculate \( x \):- For \( n = -3 \): \( x = \frac{(-3+1)\pi}{2} = -\pi \)- For \( n = -2 \): \( x = \frac{(-2+1)\pi}{2} = -\frac{\pi}{2} \)- For \( n = -1 \): \( x = \frac{(-1+1)\pi}{2} = 0 \)- For \( n = 0 \): \( x = \frac{(0+1)\pi}{2} = \frac{\pi}{2} \)- For \( n = 1 \): \( x = \pi \)

Step 6: Verify Solutions

Each \( x \) calculated falls within the interval \([-\pi, \pi]\). Graphing \( \tan(2x - \pi) \) as a reference, the zeroes should match these calculated values.

Key Concepts

Tangent Function

The tangent function, written as "\( \tan \theta \)", is a trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle. It's an essential concept in trigonometry due to its periodic nature. One key property is that the tangent function has a periodicity of \( \pi \), meaning it repeats every \( \pi \) units.

• Tangent is undefined at odd multiples of \( \frac{\pi}{2} \) because it approaches infinity at these points.
• \( \tan \theta = 0 \) whenever \( \theta = n\pi \), where \( n \) is an integer.

The function's graph is also unique, as it features vertical asymptotes at these points of undefined values, causing the graph to sharply transition. Understanding the tangent function's properties is crucial for solving equations that involve it, such as the one given in the exercise.

Interval Notation

Interval notation is a concise way to describe subsets of the real number line. It is used to specify where solutions to equations, like trigonometric ones, must lie. In our exercise, the interval \([-\pi, \pi]\) indicates all the valid \( x \)-values must be between -\( \pi \) and \( \pi \), inclusive.

• Brackets \([ \text{and} ]\) indicate inclusive boundaries, meaning the endpoints are part of the interval.
• Parentheses \(( \text{and} )\) would indicate exclusive boundaries, excluding the endpoints.

Interval notation is beneficial because it clearly denotes the range within which solutions must exist, allowing us to determine valid values during problem-solving. It also helps in visualizing the domain and range of functions, especially when combined with graphing techniques.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are true for every possible value of the involved variables. They simplify the process of solving and manipulating expressions or equations involving trigonometric terms. For instance, recognizing that \( \tan \theta = 0 \) when \( \theta = n\pi \) greatly simplifies solving the given equation.

• The Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), are crucial for transforming expressions.
• Double angle formulas, like \( \tan(2\theta) = \frac{2\tan \theta}{1-\tan^2\theta} \), are useful for rewriting and solving equations.

Using trigonometric identities effectively can lead to quicker and more efficient problem-solving methods in exercises involving complex trigonometric expressions.

Graphing

Graphing is a visual method for understanding and solving mathematical functions. In the context of trigonometric equations, graphing can help locate solutions by identifying where the graph intersects with specific values, such as zero.

• Creating a graph of \( y = \tan(2x - \pi) \) involves plotting the curve over the desired interval, \([-\pi, \pi]\).
• By evaluating where the curve hits zero, one can determine the solutions to the trigonometric equation.

Graphing is especially helpful because it allows you to visually confirm solutions obtained through algebraic manipulation, providing an additional layer of verification. By mapping the behavior of the function across different intervals, students can gain deeper insights into how the function behaves across its domain.