Solving Tan(x) - 2sin(x)tan(x) = 0 Exact Solutions On [0, 2π)
Introduction
In the realm of trigonometry, solving equations is a fundamental skill. This article delves into the process of finding exact solutions for the trigonometric equation tan(x) - 2sin(x)tan(x) = 0 within the interval [0, 2π). This range represents one complete revolution around the unit circle, making it a common domain for trigonometric problems. We'll explore the steps involved in isolating the variable x, considering trigonometric identities, and identifying all solutions within the specified interval. By understanding these techniques, you'll be better equipped to tackle a variety of trigonometric equations.
Problem Statement
Our primary goal is to determine all values of x that satisfy the equation:
tan(x) - 2sin(x)tan(x) = 0
where x lies within the interval [0, 2π). This means we're looking for solutions within one full rotation around the unit circle, starting from 0 radians and ending just before 2π radians. This interval is crucial because trigonometric functions are periodic, meaning their values repeat after a certain interval. Restricting the domain to [0, 2π) ensures we find a unique set of solutions for a single period.
To solve this equation, we'll utilize algebraic manipulation and trigonometric identities. The key is to isolate the variable x and identify the angles within the given interval that satisfy the equation. This involves understanding the relationships between trigonometric functions and their values on the unit circle. We'll also pay close attention to potential extraneous solutions that might arise from algebraic manipulations.
Step-by-Step Solution
Let's embark on the journey of finding the exact solutions. This involves a systematic approach, combining algebraic manipulations and trigonometric identities.
1. Factoring out tan(x)
Our initial equation is:
tan(x) - 2sin(x)tan(x) = 0
The first step towards solving this equation involves recognizing a common factor. Observe that tan(x) appears in both terms on the left-hand side. We can factor out tan(x), which simplifies the equation and allows us to analyze it more effectively:
tan(x) [1 - 2sin(x)] = 0
This factored form is crucial because it transforms the equation into a product of two factors equaling zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This principle is fundamental to solving many algebraic equations, including trigonometric equations.
By factoring, we've effectively broken down the original equation into two simpler equations:
- tan(x) = 0
- 1 - 2sin(x) = 0
We can now solve each of these equations independently and then combine the solutions to find the complete solution set for the original equation. This strategy of factoring and applying the zero-product property is a powerful tool in solving a wide range of mathematical problems.
2. Solving tan(x) = 0
Now, let's tackle the first of our simplified equations:
tan(x) = 0
Recall that tan(x) is defined as the ratio of sin(x) to cos(x):
tan(x) = sin(x) / cos(x)
Therefore, tan(x) will be zero when sin(x) is zero, provided that cos(x) is not also zero at the same angle (as this would result in an undefined expression). To find the solutions for tan(x) = 0 within the interval [0, 2π), we need to identify the angles where sin(x) = 0.
On the unit circle, the sine function corresponds to the y-coordinate of a point on the circle. The y-coordinate is zero at two points within the interval [0, 2π): at 0 radians and at π radians. At 2π, the y-coordinate is also 0, but 2π is not included in the interval.
Therefore, the solutions to tan(x) = 0 within the interval [0, 2π) are:
- x = 0
- x = π
These two angles represent the points where the terminal side of the angle intersects the x-axis on the unit circle. We've found our first set of solutions for the original equation.
3. Solving 1 - 2sin(x) = 0
Next, we turn our attention to the second simplified equation:
1 - 2sin(x) = 0
To solve for sin(x), we'll perform some algebraic manipulation. First, add 2sin(x) to both sides of the equation:
1 = 2sin(x)
Then, divide both sides by 2:
sin(x) = 1/2
Now, we need to find the angles within the interval [0, 2π) where the sine function equals 1/2. Recall that the sine function corresponds to the y-coordinate on the unit circle. The y-coordinate is 1/2 at two points in the interval [0, 2π). These points correspond to angles in the first and second quadrants where the reference angle is π/6 (30 degrees).
The angles are:
- x = π/6 (in the first quadrant)
- x = 5π/6 (in the second quadrant)
To understand why these are the solutions, visualize the unit circle. In the first quadrant, an angle of π/6 has a y-coordinate of 1/2. In the second quadrant, the angle 5π/6 has the same reference angle (π/6) with respect to the negative x-axis, and thus also has a y-coordinate of 1/2. These are the only two angles within the interval [0, 2π) where the sine function equals 1/2.
4. Combining Solutions
We've now solved both simplified equations that resulted from factoring the original equation. Let's gather all the solutions we've found within the interval [0, 2π):
From tan(x) = 0, we obtained:
- x = 0
- x = π
From 1 - 2sin(x) = 0, which led to sin(x) = 1/2, we found:
- x = π/6
- x = 5π/6
Therefore, the complete set of solutions for the equation tan(x) - 2sin(x)tan(x) = 0 within the interval [0, 2π) is:
x = 0, π/6, 5π/6, π
These four angles are the exact solutions to the given trigonometric equation within the specified domain. It's essential to present the solutions in a clear and organized manner, as we have done here, to ensure accuracy and understanding.
Conclusion
In this article, we successfully navigated the process of solving the trigonometric equation tan(x) - 2sin(x)tan(x) = 0 within the interval [0, 2π). By employing the techniques of factoring, applying the zero-product property, and utilizing our understanding of the unit circle and trigonometric identities, we were able to determine the exact solutions. This journey highlights the importance of a systematic approach to solving trigonometric equations.
The key takeaways from this exploration are:
- Factoring can simplify complex trigonometric equations by breaking them down into smaller, more manageable parts.
- The zero-product property is a powerful tool for solving equations in factored form.
- A strong understanding of the unit circle and the values of trigonometric functions at key angles is crucial for finding solutions.
- Paying attention to the specified interval is essential for identifying all solutions within the given domain.
The solutions we found, x = 0, π/6, 5π/6, π, represent the angles within one full rotation of the unit circle that satisfy the original equation. These solutions demonstrate the interplay between algebra and trigonometry in solving mathematical problems. By mastering these techniques, you can confidently tackle a wide array of trigonometric challenges and deepen your understanding of mathematical principles.
This approach can be generalized to solve other trigonometric equations. Always look for opportunities to factor, utilize trigonometric identities, and carefully consider the domain to find all exact solutions. Remember to check your solutions to ensure they are valid and not extraneous.
By consistently practicing and applying these methods, you'll develop proficiency in solving trigonometric equations and build a solid foundation in trigonometry. This knowledge is invaluable in various fields, including physics, engineering, and computer science, where trigonometric functions play a vital role in modeling periodic phenomena and solving complex problems.
Final Answer
0, π/6, 5π/6, π
