Problem 22 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$3 \tan ^{3} x=\tan x$$

Step-by-Step Solution

Answer

The solutions are \( x = n\pi \), \( x = \frac{\pi}{6} + n\pi \), and \( x = \frac{5\pi}{6} + n\pi \) for any integer \( n \).

Step 1: Identify the given equation

The given equation to solve is \( 3 \tan^3 x = \tan x \). Our task is to find all possible values for \( x \) that satisfy this equation.

Step 2: Simplify the equation

To simplify the equation, first factor out \( \tan x \): \( \tan x (3 \tan^2 x - 1) = 0 \). This results in two possible equations: \( \tan x = 0 \) and \( 3 \tan^2 x - 1 = 0 \).

Step 3: Solve the first equation

Solving \( \tan x = 0 \), we recognize that \( x = n\pi \) where \( n \) is an integer, because the tangent function is equal to zero for these values.

Step 4: Solve the second equation

Next, solve \( 3 \tan^2 x - 1 = 0 \). Rearrange to get \( 3 \tan^2 x = 1 \), leading to \( \tan^2 x = \frac{1}{3} \). Then, \( \tan x = \pm \frac{1}{\sqrt{3}} \).

Step 5: Find solutions for the second equation

The solutions for \( \tan x = \frac{1}{\sqrt{3}} \) are \( x = \frac{\pi}{6} + n\pi \), and the solutions for \( \tan x = -\frac{1}{\sqrt{3}} \) are \( x = \frac{5\pi}{6} + n\pi \), where \( n \) is an integer.

Step 6: Combine all solutions

Combine the solutions from both equations: \( x = n\pi \), \( x = \frac{\pi}{6} + n\pi \), and \( x = \frac{5\pi}{6} + n\pi \), where \( n \) is any integer.

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan x \), is a fundamental concept in trigonometry. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle. One of the key aspects of the tangent function is its periodicity and range. The tangent of an angle repeats every \( \pi \), meaning the function is periodic with a period of \( \pi \). This property is crucial in solving trigonometric equations as it implies that the solutions repeat in regular intervals.

Additionally, the tangent function can take on any real number, which contrasts to sine and cosine functions that are limited to a range between -1 and 1. Because of this, solving equations that involve the tangent function usually provides an infinite set of solutions, reflected by the use of "\( n\pi \)" in solutions, where \( n \) is any integer. Understanding these properties can help in visualizing and solving problems effectively when working with tangent functions.

Trigonometric Identities

Trigonometric identities are equations that hold true for every value of the variables involved. They are incredibly useful for transforming and simplifying complex expressions into more manageable forms. In the context of this exercise, recognizing the expression \( 3 \tan^3 x = \tan x \) can be simplified using identities.

Firstly, factor the equation by using a simple algebraic identity, where you have a common factor. In the equation, \( \tan x \) is common on both sides, allowing us to factor the equation as \( \tan x (3 \tan^2 x - 1) = 0 \). This reveals two simpler equations to solve: \( \tan x = 0 \) and \( 3 \tan^2 x - 1 = 0 \).

Understanding and recognizing these identities help you to deconstruct what seems like a complex equation into simpler terms, making the solution process more efficient. This approach highlights the power of trigonometric identities in solving equations.

Equation Solving Steps

The process of solving trigonometric equations often involves a systematic approach to identify and simplify the situation at hand. In this particular example, solving the equation \( 3 \tan^3 x = \tan x \), involves several key steps.

Start by identifying and understanding the equation. Factoring is the next crucial step, where you extract common terms. Here, \( \tan x \) is factored out to give \( \tan x (3 \tan^2 x - 1) = 0 \). Subsequently, split the problem into two separate equations: \( \tan x = 0 \) and \( 3 \tan^2 x - 1 = 0 \).

Each of these equations is handled independently:

• For \( \tan x = 0 \), you find that \( x = n\pi \), capturing its periodicity.
• For \( 3 \tan^2 x - 1 = 0 \), rearrange and solve to find \( \tan x = \pm \frac{1}{\sqrt{3}} \), giving solutions \( x = \frac{\pi}{6} + n\pi \) and \( x = \frac{5\pi}{6} + n\pi \).

Finally, you must combine all these solutions ensuring you capture every scenario where the original equation holds true. This systematic breakdown ensures clarity and successfully finds all valid solutions.