Problem 43 - Algebra and Trigonometry

Question

Find all solutions of the equation in the interval \([0,2 \pi).\) $$2 \sin x \tan x-\tan x=1-2 \sin x$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\).

Step 1: Simplify the Equation

Start by factoring the left side of the given equation: \(2 \sin x \tan x - \tan x = \tan x (2 \sin x - 1)\). The equation becomes: \(\tan x (2 \sin x - 1) = 1 - 2 \sin x\).

Step 2: Rearrange the Equation

Notice that the equation \(\tan x (2 \sin x - 1) = 1 - 2 \sin x\) can be rewritten to express in terms of \(\sin x\). Rearrange it by subtracting \(1 - 2 \sin x\) from both sides: \[\tan x (2 \sin x - 1) - (1 - 2 \sin x) = 0.\] This simplifies to \[\tan x = \frac{1 - 2 \sin x}{2 \sin x - 1}.\]

Step 3: Analyze Sin and Tan

Notice that the expression suggests finding values where \(\sin x\) and \(\tan x\) are related in a simple way. Consider equations like \(\tan x = \sin x\). Since \(\tan x = \frac{\sin x}{\cos x}\), set \(\sin x = \cos x\), which happens when \(x = \frac{\pi}{4} \) or \(x = \frac{5\pi}{4}\). Check these values against the original equation.

Step 4: Find Solutions

After substituting, when \(x = \frac{\pi}{4}\), check if it satisfies the original equation:\[2 \sin \frac{\pi}{4} \tan \frac{\pi}{4} - \tan \frac{\pi}{4} = 1 - 2 \sin \frac{\pi}{4},\]which simplifies to \\(1 - 1 = 1 - 1\), true. Thus, \(x = \frac{\pi}{4}\) is a solution.Similarly, check \(x = \frac{5\pi}{4}\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin x \), is a fundamental trigonometric function. It represents the y-coordinate of a point on the unit circle corresponding to a given angle \( x \). The sine function is periodic with a period of \( 2 \pi \), which means its values repeat every \( 2 \pi \) radians.

Key characteristics of the sine function include:

• Range: The sine function always yields results between \(-1\) and \(1\).
• Intercepts: \(\sin x = 0\) at integer multiples of \(\pi\), such as \(0\), \(\pi\), and \(2\pi\).
• Symmetry: \(\sin x\) is an odd function, implying that \(\sin(-x) = -\sin x\). This symmetry creates a mirror effect over the origin.
• Graph Shape: It has a smooth, wave-like pattern with regular peaks and troughs.

In our exercise, the sine function plays a crucial role because the equation is expressed in terms of both \(\sin x\) and \(\tan x\), creating a link we must explore further to find solutions.

Tangent Function

The tangent function, written as \( \tan x \), is another essential trigonometric function defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). This relationship between the sine and the cosine functions makes understanding \( \tan x \) deeply connected to these two functions.

Tangent function characteristics include:

• Periodicity: It has a period of \( \pi \), meaning \( \tan(x + \pi) = \tan x \).
• Vertical Asymptotes: Occur at odd multiples of \( \frac{\pi}{2} \), like \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc., where the function is undefined because \( \cos x \) is zero.
• Range: Unlike the sine function, \( \tan x \) can take any real number as it ranges from \(-\infty\) to \(+\infty\).
• Symmetry: \(\tan x\) is an odd function too, which means \(\tan(-x) = -\tan x\).

The equation in our exercise involves rearranging its expressions to relate \( \sin x \) and \( \tan x \), underscoring the importance of these relationships in trigonometric analysis.

Solution of Equations

Solving trigonometric equations involves finding all angle values that satisfy the equation within a specific interval. In our problem, we are tasked with finding values within the interval \([0, 2\pi)\). This involves considering both the periodic nature of trigonometric functions and their specific properties.

Here are some critical steps when dealing with such problems:

• Simplification: Start by simplifying the equation, factor common terms, and rewrite expressions using known identities.
• Rearrangement: Align the equation in terms of a single trigonometric function when possible to isolate variables. In our example, this involved expressing everything in terms of \(\sin x\).
• Analysis: Determine specific angle solutions that satisfy rearranged and factored equations. This may involve setting equations like \( \sin x = \cos x \) as seen in this exercise, where these conditions need checking.
• Verification: Always verify possible solutions against the original equation to ensure accuracy. Missteps can occur if assumptions include angles not meeting all given conditions.

In this scenario, achieving solutions involved identifying known angles, such as \( x = \frac{\pi}{4} \), where these conditions hold true under the problem's restrictions.