Solving Trigonometric Equations Exact Solutions Of -3tan²x + 1 = 0 In Radians
Introduction: Unveiling the Solutions of Trigonometric Equations
In the realm of mathematics, trigonometric equations hold a significant place, particularly in fields like physics, engineering, and computer science. These equations involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. Finding the solutions to these equations is a fundamental skill, often requiring a blend of algebraic manipulation and trigonometric identities. This comprehensive guide delves into the process of solving the trigonometric equation -3tan²x + 1 = 0, providing a step-by-step approach to arrive at the exact solutions in radians. Understanding these solutions is crucial for various applications, ranging from modeling periodic phenomena to analyzing wave behavior. The ability to solve trigonometric equations efficiently and accurately is an invaluable asset for any student or professional in STEM fields. So, let's embark on this journey to master the art of solving trigonometric equations and unravel the mysteries they hold. We will explore the underlying principles, employ essential techniques, and gain a profound understanding of how to obtain exact solutions. This knowledge will not only empower you to tackle similar problems but also enhance your overall mathematical proficiency. Let's dive in and unlock the world of trigonometric equations together!
Problem Statement: Decoding -3tan²x + 1 = 0
The specific trigonometric equation we aim to solve is -3tan²x + 1 = 0. This equation involves the tangent function, a cornerstone of trigonometry. Our goal is to find all the values of x, expressed in radians, that satisfy this equation. These values represent the angles for which the tangent function, when squared and manipulated as in the equation, results in zero. To achieve this, we will employ a combination of algebraic techniques and trigonometric principles. First, we will isolate the tan²x term, followed by taking the square root to find the possible values of tan x. Then, we will utilize the properties of the tangent function and the unit circle to determine the angles x that correspond to these values. Finally, we will express the general solutions, accounting for the periodic nature of the tangent function. This step-by-step approach will ensure that we cover all possible solutions and understand the underlying concepts thoroughly. Solving this equation not only provides the answer but also reinforces our understanding of trigonometric functions and their behavior. It highlights the importance of algebraic manipulation in simplifying complex equations and the application of trigonometric identities in finding solutions. So, let's proceed with the solution process and unravel the angles that satisfy this equation.
Step-by-Step Solution: Solving for x in Radians
To solve the equation -3tan²x + 1 = 0, we follow these steps:
Step 1: Isolate tan²x
Begin by isolating the tan²x term. Add 3tan²x to both sides of the equation:
1 = 3tan²x
Then, divide both sides by 3:
tan²x = 1/3
This step is crucial as it simplifies the equation and allows us to focus on the tangent function. By isolating tan²x, we prepare the equation for the next step, which involves taking the square root. This algebraic manipulation is a common technique in solving various types of equations, and mastering it is essential for mathematical problem-solving. The isolation of tan²x not only simplifies the equation but also provides a clear path towards finding the solutions. It sets the stage for applying trigonometric principles and understanding the behavior of the tangent function. This initial step is a building block for the subsequent steps, and a solid understanding of this process is vital for tackling more complex trigonometric equations. Let's proceed to the next step, where we will take the square root of both sides and further simplify the equation.
Step 2: Take the Square Root
Take the square root of both sides of the equation:
tan x = ±√(1/3)
This simplifies to:
tan x = ±1/√3
Taking the square root introduces both positive and negative solutions, which is a critical step in ensuring we find all possible values of x. The ± sign indicates that there are two sets of solutions to consider, one where tan x is positive and another where tan x is negative. This step highlights the importance of considering all possibilities when solving equations involving squares. By taking the square root, we reduce the equation to a simpler form that directly relates tan x to a numerical value. This allows us to use our knowledge of the tangent function and its values at specific angles to find the corresponding solutions for x. The introduction of the ± sign is a key concept in solving equations of this nature, and understanding its significance is crucial for obtaining complete and accurate solutions. Now, let's move on to the next step, where we will determine the angles that satisfy these tangent values.
Step 3: Find Reference Angles
The reference angles are the angles in the first quadrant that have the same trigonometric function value (ignoring the sign). We know that tan(π/6) = 1/√3. Therefore, the reference angle is π/6.
Reference angles are essential tools in solving trigonometric equations. They provide a basis for finding all solutions within a given period. By identifying the reference angle, we can use the properties of trigonometric functions in different quadrants to determine all angles that satisfy the equation. The tangent function, in particular, has a period of π, which means its values repeat every π radians. Understanding reference angles and periodicity is crucial for expressing the general solutions of trigonometric equations. In this case, the reference angle of π/6 serves as a foundation for finding all angles where the tangent function has a value of ±1/√3. By considering the quadrants where tangent is positive and negative, we can identify all the angles that satisfy the equation within one period. This step is a cornerstone of the solution process, and a clear understanding of reference angles is vital for accurate results. Let's proceed to the next step, where we will find the angles in all quadrants that correspond to the reference angle.
Step 4: Determine Angles in All Quadrants
The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants.
For tan x = 1/√3:
In the first quadrant, x = π/6.
In the third quadrant, x = π + π/6 = 7π/6.
For tan x = -1/√3:
In the second quadrant, x = π - π/6 = 5π/6.
In the fourth quadrant, x = 2π - π/6 = 11π/6.
This step is critical for finding all the solutions within the interval [0, 2π). By considering the signs of the tangent function in different quadrants, we can identify all the angles that have a tangent value of ±1/√3. The first quadrant angle, π/6, serves as the reference angle, and we use it to find the corresponding angles in the other quadrants. In the third quadrant, where tangent is positive, we add π to the reference angle. In the second and fourth quadrants, where tangent is negative, we subtract the reference angle from π and 2π, respectively. This systematic approach ensures that we capture all possible solutions within one period of the tangent function. Understanding the quadrant signs of trigonometric functions is a fundamental concept in trigonometry, and this step reinforces that understanding. By carefully considering each quadrant, we can accurately determine all the angles that satisfy the given equation. Let's proceed to the final step, where we will express the general solutions, accounting for the periodicity of the tangent function.
Step 5: Express General Solutions
Since the tangent function has a period of π, we can express the general solutions by adding kπ to each solution, where k is an integer.
x = π/6 + kπ
x = 5π/6 + kπ
These are the general solutions to the equation -3tan²x + 1 = 0.
Expressing the general solutions is the final step in solving trigonometric equations. It involves accounting for the periodic nature of trigonometric functions. The tangent function, in particular, has a period of π, meaning its values repeat every π radians. Therefore, we add kπ to each solution to represent all possible solutions, where k is any integer. This notation ensures that we capture all angles that satisfy the equation, regardless of how many full rotations around the unit circle they represent. The general solutions provide a complete and concise representation of all possible solutions. Understanding periodicity is a key concept in trigonometry, and this step reinforces that understanding. By expressing the general solutions, we demonstrate a comprehensive understanding of the equation and its solutions. This final step completes the solution process, providing a clear and accurate answer to the problem. The general solutions x = π/6 + kπ and x = 5π/6 + kπ represent all the angles that satisfy the equation -3tan²x + 1 = 0. Now, let's summarize the solution and highlight the key steps involved.
Summary of the Solution: Recap of Key Steps
In summary, the exact solutions of the equation -3tan²x + 1 = 0 are:
x = π/6 + kπ
x = 5π/6 + kπ
Where k is an integer.
We arrived at these solutions by following a systematic approach:
- Isolating tan²x: We began by isolating the tan²x term, simplifying the equation and making it easier to solve.
- Taking the Square Root: Taking the square root of both sides introduced both positive and negative solutions, which is crucial for finding all possible values of x.
- Finding Reference Angles: We identified the reference angle, which is the angle in the first quadrant that has the same trigonometric function value (ignoring the sign).
- Determining Angles in All Quadrants: We considered the signs of the tangent function in different quadrants to find all the angles that have a tangent value of ±1/√3.
- Expressing General Solutions: Finally, we expressed the general solutions by adding kπ to each solution, where k is an integer, accounting for the periodicity of the tangent function.
This step-by-step process highlights the importance of algebraic manipulation, trigonometric identities, and understanding the properties of trigonometric functions. By following these steps, we can confidently solve a wide range of trigonometric equations. The solutions we obtained represent all the angles that satisfy the given equation, providing a complete and accurate answer. This summary serves as a valuable recap of the solution process, reinforcing the key concepts and techniques involved. Now, let's conclude with a final thought on the significance of solving trigonometric equations.
Conclusion: Mastering Trigonometric Solutions
Solving trigonometric equations like -3tan²x + 1 = 0 is a fundamental skill in mathematics. The solutions we found, x = π/6 + kπ and x = 5π/6 + kπ, represent a family of angles that satisfy the given equation. Mastering these techniques is essential for various applications in mathematics, physics, engineering, and other STEM fields. The ability to solve trigonometric equations allows us to model periodic phenomena, analyze wave behavior, and solve a wide range of problems involving angles and triangles. This comprehensive guide has provided a step-by-step approach to solving this specific equation, but the principles and techniques discussed can be applied to a variety of trigonometric equations. By understanding the properties of trigonometric functions, using algebraic manipulation, and considering the periodicity of the functions, we can confidently tackle these equations and find their solutions. The journey of solving trigonometric equations is not just about finding the answers; it's about developing a deeper understanding of mathematical concepts and enhancing problem-solving skills. So, embrace the challenge, practice regularly, and continue to explore the fascinating world of trigonometry.
