Problem 41 - Precalculus with Limits

Question

In Exercises 39-44, solve the multiple-angle equation. \( \tan 3x = 1 \)

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the equation \( \tan 3x = 1\) are \(x = \frac{\pi}{12} + \frac{n\pi}{3}\) and \(x = \frac{5\pi}{12} + \frac{n\pi}{3}\), where \(n\) is an integer.

Step 1: Solve for the angle where tangent is 1

The angle \(a\) where \( \tan(a) = 1\) can be found on the unit circle as \(a = \frac{\pi}{4}\) and \(a = \frac{5\pi}{4}\), since tangent of these angles is 1. This is within one period of the tangent function, which is \(\pi\). All angles are covered by adding \(n\pi\) to each one of these solutions, \(n\) being an integer.

Step 2: Solve for \(3x\)

The original equation is \( \tan 3x = 1\), so those solutions found should equal to \(3x\) instead of \(x\). This means \(3x = \frac{\pi}{4} + n\pi\) and \(3x = \frac{5\pi}{4} + n\pi\).

Step 3: Solve for \(x\)

The last step is to solve for \(x\) so dividing each equation by 3 gives us the solutions: \(x = \frac{\pi}{12} + \frac{n\pi}{3}\) and \(x = \frac{5\pi}{12} + \frac{n\pi}{3}\). Each one of these solutions accounts for all values of \(x\) where \( \tan 3x = 1\).

Key Concepts

Multiple-Angle Equations

Multiple-angle equations involve trigonometric functions with angles that are multiples of a variable, such as \( \tan 3x \). Solving these types of equations requires us to consider the trigonometric properties of the angle and then isolate the variable. This generally involves multiple steps, culminating in a solution set that encompasses all possible solutions based on the periodicity of the trigonometric function involved.

In our case, we need to solve \( \tan 3x = 1 \). This means we first find angles where the tangent equals 1, and then divide by the multiple to find the general solution. Multiple-angle equations often result in solutions that need to be expanded to accommodate the period of the trigonometric function.

By recognizing these angles and applying the periodicity, we see that each solution should include \( n\pi \), where \( n \) is an integer. Once these angles have been identified, solving the equation means systematically working through these angles to solve for the variable in terms of \( x \).

Unit Circle

The unit circle is a valuable tool when solving trigonometric equations because it visualizes trigonometric functions as points on a circle. With a radius of 1, this circle helps us determine the sine, cosine, and tangent of common angles, making it easier to solve equations like \( \tan 3x = 1 \).

On the unit circle, the tangent function corresponds to the slope of the angle from the circle's center to the circle's edge. The equation \( \tan 3x = 1 \) asks for angles where this slope is 1. In the first cycle of the unit circle, these commonly known angles are \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \).

By knowing these unit circle values, we can extend our solutions by considering the periodic nature of the tan function, allowing us to find all possible solutions for \( 3x \) and consequently for \( x \) within the range of angles. Identifying angles like these on the unit circle streamlines the process of solving trigonometric equations.

Periodicity of Tangent Function

The tangent function is unique in that it repeats itself every \( \pi \) radians. This property, called periodicity, is crucial when solving equations involving tangent. The equation \( \tan 3x = 1 \) utilizes this property by acknowledging that any solution can be repeated after adding \( n\pi \), where \( n \) is an integer.

Given the known angles where \( \tan \theta = 1 \) are \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \), the periodicity helps generate a series of solutions: \( \frac{\pi}{4} + n\pi \), and \( \frac{5\pi}{4} + n\pi \). This happens because tan has undefined values at each vertical asymptote, such as at \( \frac{\pi}{2} \), which are avoided by using half a period to oscillate to the opposite side.

Understanding the periodicity concept is essential for solving serious equations like \( \tan 3x = 1 \), as it explains where solutions converge and when they reoccur. Format your answers to include these periodic jumps to ensure you've captured all possible values for \( x \).