Problem 39 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation that are in the interval \([0,2 \pi).\) \(\tan 2 x=\tan x\)

Step-by-Step Solution

Answer

The solutions in the interval \([0, 2\pi)\) are \(x = 0\) and \(x = \pi\).

Step 1: Apply the Tangent Identity

Use the identity \( \tan A - \tan B = \frac{\sin(A-B)}{\cos A \cos B} \) to express \( \tan 2x - \tan x = 0 \) as \( \sin(2x - x) = 0 \). This simplifies to \( \sin x = 0 \).

Step 2: Solve \( \sin x = 0 \)

The solutions to \( \sin x = 0 \) in the interval \([0, 2\pi)\) are \( x = 0, \pi \).

Step 3: Use the Tangent Double Angle Identity

Rewrite \( \tan 2x = \tan x \) using the identity \( \tan 2x = \frac{2\tan x}{1-\tan^2 x} = \tan x \). This gives the equation \( \frac{2\tan x}{1-\tan^2 x} = \tan x \).

Step 4: Simplify the New Equation

Multiply both sides of \( \frac{2\tan x}{1-\tan^2 x} = \tan x \) by \( 1-\tan^2 x \) to get \( 2\tan x = \tan x - \tan^3 x \) which simplifies to \( \tan x (1 + \tan^2 x) = 0 \).

Step 5: Find Additional Solutions from Simplified Equation

The equation \( \tan x (1 + \tan^2 x) = 0 \) provides solutions when \( \tan x = 0 \) or \( 1 + \tan^2 x = 0 \). \( \tan x = 0 \) provides \( x = 0, \pi \). The equation \( 1 + \tan^2 x = 0 \) has no real solution as \( \tan^2 x \) cannot be negative.

Key Concepts

tangent identity

Understanding the tangent identity is essential when solving trigonometric equations involving tangent functions. A basic but crucial identity is the difference of tangents identity:

• \( \tan A - \tan B = \frac{\sin(A-B)}{\cos A \cos B} \).

This identity shows the connection between tangent and sine functions, offering a pathway to simplify complex expressions. In the equation \( \tan 2x = \tan x \), applying the identity helps us transform it into \( \sin(2x - x) = 0 \), or simply \( \sin x = 0 \). This transformation is key because solving for \( \sin x = 0 \) is often more straightforward than dealing directly with tangents.
Practicing the use of tangent identities allows us to navigate complex trigonometric problems with more ease.

sine function

The sine function is one of the fundamental trigonometric functions. It provides the vertical distance from a point on a unit circle to the x-axis. The equation \( \sin x = 0 \) is especially significant since it simplifies trigonometric equations like the one we discussed. Solving \( \sin x = 0 \) within the interval \([0, 2\pi)\) yields important solutions:

• \( x = 0 \)
• \( x = \pi \)

These solutions arise from the points where the sine function intersects the x-axis in a single revolution of the unit circle. Understanding where the sine function crosses the x-axis helps in solving not only simple trigonometric equations but also those involving combined or chained trigonometric concepts. The simplicity and periodic nature of the sine function make it a cornerstone in trigonometry, and knowing its properties is essential for success.

tangent double angle identity

The tangent double angle identity provides a useful equation for expressions involving the tangent of twice an angle:

• \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \)

This identity allows us to express \( \tan 2x \) entirely in terms of \( \tan x \). It's particularly helpful in equations like \( \tan 2x = \tan x \), as it transforms the equation into something more solvable by manipulating algebraic expressions.
For instance, multiplying both sides of \( \frac{2\tan x}{1-\tan^2 x} = \tan x \) by \( 1-\tan^2 x \) gives the equation \( 2\tan x = \tan x - \tan^3 x \) which simplifies to \( \tan x (1 + \tan^2 x) = 0 \).
This step effectively helps find additional solutions by isolating \( \tan x \) and solving for situations when it equals zero or explores conditions under which other terms vanish. Understanding this identity aids in tackling more advanced trigonometric equations by breaking them into manageable parts.