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Magazine Chapter 8: Problem 43
Find all solutions of the equation in the interval \([0,2 \pi).\) $$2 \sin x \tan x-\tan x=1-2 \sin x$$
Short Answer
Expert verified
The solutions are \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\).
Step by step solution
01
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: \(\tan x (2 \sin x - 1) = 1 - 2 \sin x\).
02
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 \[\tan x = \frac{1 - 2 \sin x}{2 \sin x - 1}.\]
03
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.
04
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 \(x = \frac{5\pi}{4}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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 \(\tan(-x) = -\tan x\).
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:
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.
