Problem 38 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\sin 2 x=2 \tan 2 x$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{n\pi}{2}\) for any integer \(n\).

Step 1: Rewrite the Given Equation Using Trigonometric Identities

We start by rewriting \(\sin 2x = 2 \tan 2x\) using trigonometric identities. We know that \(\tan 2x = \frac{\sin 2x}{\cos 2x}\). Substituting this into the equation gives: \[ \sin 2x = 2 \cdot \frac{\sin 2x}{\cos 2x} \] Multiplying both sides by \(\cos 2x\) to eliminate the fraction results in:\[ \sin 2x \cdot \cos 2x = 2 \sin 2x \]

Step 2: Simplify the Equation

We factor out \(\sin 2x\) from the equation:\[ \sin 2x (\cos 2x - 2) = 0 \]This implies that either \(\sin 2x = 0 \) or \(\cos 2x - 2 = 0\). However, \(\cos 2x\) must satisfy \(-1 \leq \cos 2x \leq 1\), so \(\cos 2x - 2 = 0\) has no solution. Thus, we only need to consider the case \(\sin 2x = 0\).

Step 3: Solve for \(x\) When \(\sin 2x = 0\)

The equation \(\sin 2x = 0\) holds when:\[ 2x = n\pi \]where \(n\) is an integer. Solving for \(x\) gives:\[ x = \frac{n\pi}{2} \]

Step 4: Define the General Solution

Since \(n\) can be any integer, the general solution for the equation is:\[ x = \frac{n\pi}{2} \quad \text{for integers } n \]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometric equations. These identities help us transform complex expressions into more manageable forms. In the original exercise, we used the identity for the tangent function, which is expressed as

• \( \tan 2x = \frac{\sin 2x}{\cos 2x} \)

This identity allows us to express tangent in terms of sine and cosine, making it easier to manipulate the equation. Applying trigonometric identities often simplifies the equation and reveals solutions that aren't immediately obvious. By substituting \( \tan 2x \) in the given equation, we could clear out fractions and focus on essential trigonometric functions like sine and cosine.

Sin Function

The sine function is one of the fundamental trigonometric functions and is denoted as \( \sin \). It is essential in this problem as a part of both the given and transformed equations.

• The sine function is periodic with a period of \( 2\pi \).
• Its values range from -1 to 1.

In our equation, \( \sin 2x \) plays a crucial role. When we simplify the equation to \( \sin 2x (\cos 2x - 2) = 0 \), it leads us to focus on the solutions where the sine function equals zero. This results in the simple condition \( \sin 2x = 0 \). Understanding how sine behaves over its cycle helps us find where it intersects the x-axis, i.e., when it equals zero.

Tan Function

The tangent function, noted as \( \tan \), can be more complex due to its unique properties. It is the ratio of sine and cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)

The function is periodic with a period of \( \pi \), and unlike sine and cosine, it does not have a fixed range.

• The tangent function approaches infinity as it nears the vertical asymptotes where \( \cos x = 0 \).

In the original equation transformed to involve the tangent function, we quickly saw how eliminating the derivative \( \frac{\sin 2x}{\cos 2x} \) with the identities helped simplify our solution path by reducing it to simpler trigonometric functions.

General Solution

A general solution in trigonometry provides all possible solutions to a given equation over its periodic cycles. For trigonometric functions like sine, these solutions need to reflect the function's periodic nature.

• For \( \sin 2x = 0 \), we find that \( 2x = n\pi \), where \( n \) is an integer.

Solving for \( x \) gives us \( x = \frac{n\pi}{2} \), indicating the solutions are spaced \( \frac{\pi}{2} \) apart. The general solution allows us to identify every possible x-value that satisfies the equation, covering infinite cycles over the function's domain. Understanding the periodic nature of sine helps frame these solutions accurately, reinforcing the critical advantage of knowing the function's intervals and periods.