Problem 39 - Algebra and Trigonometry with Analytic Geometry

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 are \(x = 0\) and \(x = \pi\).

Step 1: Use Trigonometric Identity

We start by using the identity for tangent differences, \( \tan A = \tan B \Rightarrow A = B + n\pi \), where \(n\) is an integer. Given \( \tan 2x = \tan x \), we have:\[2x = x + n\pi\].

Step 2: Solve for x

Rearrange the equation from Step 1:\[2x - x = n\pi \x = n\pi\].This gives a general solution \(x = n\pi\).

Step 3: Determine Solutions Within Interval

We need solutions within the interval \([0, 2\pi)\). By substituting values of \(n\):- For \(n = 0\), \(x = 0\).- For \(n = 1\), \(x = \pi\).As \(n = 2\) results in \(x = 2\pi\), which is outside the interval, it is not valid.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, often abbreviated as "tan." It is defined as the ratio of the sine function to the cosine function, specifically \(\tan x = \frac{\sin x}{\cos x}\). The tangent function is periodic, with a period of \(\pi\), meaning \(\tan(x) = \tan(x + n\pi)\) for any integer \(n\). This periodicity is important, as it allows us to use the tangent function in various trigonometric equations and identities.
The graph of the tangent function has vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\), where the function has undefined values. This gives the graph its characteristic repeating pattern of curves spanning between each asymptote.
When solving an equation involving the tangent function, it is crucial to consider this periodic nature, especially for equations like \(\tan 2x = \tan x\) that involve the tangent of double angles.

Trigonometric Identity

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable for which the functions are defined. One key identity used in the given problem is \(\tan A = \tan B \Rightarrow A = B + n\pi\), where \(n\) is an integer. This identity states that if two tangent expressions are equal, their angles differ by an integer multiple of the tangent's period, which is \(\pi\).
This identity helps transform the problem \(\tan 2x = \tan x\) into a simpler linear form \(2x = x + n\pi\). Simplifying this using the identity, we realize that \(x = n\pi\). Essentially, solving trigonometric equations often involves rewriting them using such identities to make them easier to handle.

Interval Solutions

For any trigonometric equation, finding solutions within a specific interval requires careful analysis. Here, we need solutions within the interval \( [0, 2\pi)\). Using the identity solution \( x = n\pi \,\) we substitute integer values for \( n \) to find valid solutions within this interval.

• For \( n = 0 \,\) we find \( x = 0 \.\)
• For \( n = 1 \,\) we calculate \( x = \pi \.\)
• For \( n = 2 \,\) we get \( x = 2\pi \,\) which is not included in the interval because the interval ends at \( 2\pi \,\).

Thus, the only solutions to \(\tan 2x = \tan x\) within \( [0, 2\pi) \,\) are \( x = 0 \,\) and \( x = \pi \.\) Finding these specific interval solutions ensures we accurately address the problem's requirements for given interval constraints. In many exercises, ensuring solution validity within a defined interval is as crucial as deriving the general solution.