Problem 55 - Trigonometry

Question

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \tan ^{2} 3 x=1 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{12} + \frac{k\pi}{3}\) and \(x = \frac{\pi}{4} + \frac{k\pi}{3}\) for integer \(k\).

Step 1: Understand the Given Equation

We are given the equation \(\tan^2 3x = 1\) which means the square of the tangent of \(3x\) is equal to 1. We want to find the values of \(x\) in radians.

Step 2: Solve for \(\tan 3x\)

The equation \(\tan^2 3x = 1\) means that \(\tan 3x\) could be either \(\pm 1\). So, we need to solve two scenarios: \(\tan 3x = 1\) and \(\tan 3x = -1\).

Step 3: Solve \(\tan 3x = 1\)

The tangent function equals 1 at angles \(\frac{\pi}{4} + k\pi\) where \(k\) is an integer, because the tangent function has a period of \(\pi\). So, setting \(3x = \frac{\pi}{4} + k\pi\), solve for \(x\):\[3x = \frac{\pi}{4} + k\pi \x = \frac{\pi}{12} + \frac{k\pi}{3}\].

Step 4: Solve \(\tan 3x = -1\)

The tangent function equals -1 at angles \(\frac{3\pi}{4} + k\pi\), again recognizing the period of \(\pi\). Setting \(3x = \frac{3\pi}{4} + k\pi\), solve for \(x\):\[3x = \frac{3\pi}{4} + k\pi \x = \frac{\pi}{4} + \frac{k\pi}{3}\].

Step 5: Combine Solutions

Combine the solutions from both cases. The solutions for \(x\) are given by:\[x = \frac{\pi}{12} + \frac{k\pi}{3}\quad \text{and} \quad x = \frac{\pi}{4} + \frac{k\pi}{3}\quad \text{for integer } k.\]

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function represented as \( \tan \). It relates an angle in a right triangle to the ratio of the opposite side to the adjacent side. In the context of the unit circle, \( \tan \theta \) is defined as the ratio of \( \sin \theta \) to \( \cos \theta \), which is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This definition is crucial when analyzing angles beyond right triangles, as it helps extend the function's scope to all real numbers, except where the cosine is zero. This function can take any real value and is particularly useful because of its periodic behavior.
When solving an equation like \( \tan^2 3x = 1 \), you interpret it to find values of the angle \( 3x \) such that the tangent is either 1 or -1. This step is vital for understanding the function's behavior within the context of periodic phenomena and recurring angles. Recognizing the tangent function and its properties is essential for solving trigonometric equations effectively.

Periodicity

Periodicity is a key property of trigonometric functions, meaning they repeat values in a regular pattern over intervals. For the tangent function, periodicity means that after a specific interval, the values of the tangent function repeat. The standard period of the tangent function is \( \pi \).
This property is immensely helpful in solving equations such as \( \tan 3x = 1 \) because it provides a way to find all possible solutions. When you solve \( \tan 3x = 1 \), you find solutions within one interval and then extend these solutions by adding multiples of the function's period. Specifically, this means if \( 3x = \frac{\pi}{4} + k\pi \), then \( x \) solutions can be found by solving \( x = \frac{\pi}{12} + \frac{k\pi}{3} \), covering all values of \( x \) where \( k \) is an integer.
Understanding periodicity is crucial because it ensures you account for all possible solutions when solving trigonometric equations.

Solving Equations in Radians

In trigonometry, solving equations often involves finding angle measures in radians. The radian measure is advantageous as it aligns with the natural properties of circles and provides a more direct link to calculus topics. To solve an equation like \( \tan^2 3x = 1 \), it is crucial to determine solutions for \( x \) in radians.
Here's a simple process to follow:

• Recognize the standard angles and their radian measures. For example, \( \frac{\pi}{4} \) is a common angle where \( \tan \) is 1.
• Use the properties of the tangent's periodicity (period \( \pi \)) to express all possible solutions.
• Expand these solutions to cover the full set of angles by adding \( k\pi \), where \( k \) is any integer.

In this equation, solving for \( x \) requires manipulating the equation into the form of \( \tan 3x = 1 \) or \( \tan 3x = -1 \), and applying the tangent's period to derive exact solutions for \( x \). Working in radians ensures that solutions are precise and aligned with theoretical and practical applications involving circles and waves.