Solving 2cos³x - 3cos²x + Cosx + 2 = 0 A Trigonometric Equation
Introduction: Unveiling the Trigonometric Puzzle
In the realm of mathematics, trigonometric equations often present intricate puzzles that demand a blend of algebraic manipulation and trigonometric identities to unravel. Among these equations, polynomial expressions involving trigonometric functions stand out as a particularly engaging challenge. This article delves into the comprehensive solution of the trigonometric equation 2cos³x - 3cos²x + cosx + 2 = 0. By employing a systematic approach, we will explore the underlying algebraic structure, identify potential solutions, and navigate the intricacies of trigonometric functions to arrive at a complete understanding of the equation's roots.
This exploration is not merely an academic exercise; it holds significant implications for various fields of science and engineering. Trigonometric equations form the bedrock of numerous mathematical models that describe periodic phenomena, such as oscillations, waves, and cyclical processes. From the motion of celestial bodies to the behavior of electrical circuits, understanding trigonometric equations is crucial for accurate analysis and prediction. Moreover, the techniques employed in solving these equations serve as a foundation for tackling more complex mathematical problems, fostering analytical thinking and problem-solving skills.
Method 1: Algebraic Manipulation and Factorization
Our journey to solve the equation 2cos³x - 3cos²x + cosx + 2 = 0 begins with a crucial step: recognizing the equation's inherent algebraic structure. Observe that the equation is a cubic polynomial in terms of cos x. This recognition allows us to leverage the powerful tools of algebra, particularly factorization, to simplify the problem.
To facilitate factorization, we can introduce a substitution: let y = cos x. This substitution transforms the trigonometric equation into a purely algebraic cubic equation: 2y³ - 3y² + y + 2 = 0. Now, our focus shifts to finding the roots of this cubic equation.
Several methods exist for factoring cubic polynomials, but one effective approach is to test for rational roots. The Rational Root Theorem guides us in this process, stating that any rational root of the polynomial must be a divisor of the constant term (2) divided by a divisor of the leading coefficient (2). This narrows down the possible rational roots to ±1, ±2, ±1/2.
By direct substitution, we discover that y = -1/2 is a root of the cubic equation. This crucial finding allows us to factor out the term (y + 1/2) or, equivalently, (2y + 1) from the polynomial. Performing polynomial division or synthetic division, we obtain the factorization: (2y + 1)(y² - 2y + 2) = 0.
Now, we have reduced the cubic equation to a product of a linear factor (2y + 1) and a quadratic factor (y² - 2y + 2). Setting each factor to zero, we obtain two equations: 2y + 1 = 0 and y² - 2y + 2 = 0.
The first equation, 2y + 1 = 0, yields the solution y = -1/2. The second equation, y² - 2y + 2 = 0, is a quadratic equation. We can use the quadratic formula to find its roots: y = [2 ± √((-2)² - 4 * 1 * 2)] / (2 * 1) = [2 ± √(-4)] / 2 = 1 ± i. These roots are complex numbers, which means they do not correspond to real values of cos x since the cosine function's range is [-1, 1].
Having found the valid solution y = -1/2, we revert to our original substitution y = cos x. This gives us the trigonometric equation cos x = -1/2. Now, our task is to find the values of x that satisfy this equation.
Method 2: Solving for x and General Solutions
Having simplified the original equation to cos x = -1/2, we now embark on the crucial task of finding the solutions for x. To accomplish this, we need to delve into the fundamental properties of the cosine function and its behavior within the unit circle.
The cosine function, denoted as cos x, represents the x-coordinate of a point on the unit circle corresponding to an angle x measured in radians. The unit circle, a circle with a radius of 1 centered at the origin, provides a visual and intuitive framework for understanding trigonometric functions. The cosine function exhibits periodicity, meaning its values repeat after a fixed interval. The period of the cosine function is 2π, which corresponds to a full revolution around the unit circle.
To find the solutions for cos x = -1/2, we first identify the angles on the unit circle where the x-coordinate is equal to -1/2. These angles correspond to the points where the vertical line x = -1/2 intersects the unit circle. By recalling the special angles and their corresponding cosine values, we recognize that cos(2π/3) = -1/2 and cos(4π/3) = -1/2. These two angles, 2π/3 and 4π/3, lie within the interval [0, 2π) and represent the principal solutions of the equation.
However, due to the periodic nature of the cosine function, there are infinitely many solutions to the equation cos x = -1/2. To express the general solution, we must account for all angles that are coterminal with the principal solutions. Coterminal angles are angles that share the same terminal side on the unit circle. They differ from the principal solutions by integer multiples of the period, 2π.
Therefore, the general solution to the equation cos x = -1/2 can be expressed as:
x = 2π/3 + 2πk, where k is an integer x = 4π/3 + 2πk, where k is an integer
These two expressions encompass all possible solutions to the equation, capturing the periodic behavior of the cosine function. The integer k represents the number of full revolutions around the unit circle, allowing us to generate an infinite set of solutions.
By carefully considering the properties of the cosine function and its periodic nature, we have successfully determined the general solutions to the trigonometric equation cos x = -1/2. These solutions provide a comprehensive understanding of the equation's roots and their distribution along the real number line.
Verification of Solutions
In mathematics, verification is a crucial step in the problem-solving process, ensuring the accuracy and validity of our solutions. After obtaining the general solutions for the trigonometric equation 2cos³x - 3cos²x + cosx + 2 = 0, it is essential to verify that these solutions indeed satisfy the original equation. This process not only confirms the correctness of our calculations but also deepens our understanding of the equation's behavior.
To verify the solutions, we can substitute specific values of x from the general solutions back into the original equation. Let's consider a few examples:
- x = 2π/3 (k = 0 in the first general solution)
Substituting x = 2π/3 into the original equation, we get:
2cos³(2π/3) - 3cos²(2π/3) + cos(2π/3) + 2 = 2(-1/2)³ - 3(-1/2)² + (-1/2) + 2 = -1/4 - 3/4 - 1/2 + 2 = 0
This confirms that x = 2π/3 is indeed a solution.
- x = 4π/3 (k = 0 in the second general solution)
Substituting x = 4π/3 into the original equation, we get:
2cos³(4π/3) - 3cos²(4π/3) + cos(4π/3) + 2 = 2(-1/2)³ - 3(-1/2)² + (-1/2) + 2 = -1/4 - 3/4 - 1/2 + 2 = 0
This confirms that x = 4π/3 is also a solution.
- x = 8π/3 (k = 1 in the first general solution)
Substituting x = 8π/3 into the original equation, we get:
2cos³(8π/3) - 3cos²(8π/3) + cos(8π/3) + 2 = 2(-1/2)³ - 3(-1/2)² + (-1/2) + 2 = -1/4 - 3/4 - 1/2 + 2 = 0
This confirms that x = 8π/3 is also a solution.
These examples demonstrate that the general solutions we obtained satisfy the original equation. By substituting various values of x generated from the general solutions, we can further reinforce our confidence in the correctness of our results. This verification process underscores the importance of rigor in mathematical problem-solving.
Conclusion
In this comprehensive exploration, we have successfully unraveled the intricacies of the trigonometric equation 2cos³x - 3cos²x + cosx + 2 = 0. By employing a combination of algebraic manipulation, trigonometric identities, and a deep understanding of the cosine function's properties, we have arrived at the complete solution set.
Our journey began with the crucial step of recognizing the equation's algebraic structure as a cubic polynomial in terms of cos x. This recognition allowed us to leverage the powerful tools of algebra, particularly factorization, to simplify the problem. Through the substitution y = cos x, we transformed the trigonometric equation into a purely algebraic cubic equation, 2y³ - 3y² + y + 2 = 0.
We then skillfully applied the Rational Root Theorem to identify potential rational roots of the cubic equation, leading us to the discovery that y = -1/2 is a root. This pivotal finding enabled us to factor the cubic polynomial, reducing it to a product of a linear factor (2y + 1) and a quadratic factor (y² - 2y + 2). By setting each factor to zero, we obtained two equations: 2y + 1 = 0 and y² - 2y + 2 = 0.
The linear equation yielded the solution y = -1/2, while the quadratic equation produced complex roots that did not correspond to real values of cos x. Reverting to our original substitution, we arrived at the trigonometric equation cos x = -1/2.
To solve this equation, we delved into the fundamental properties of the cosine function and its behavior within the unit circle. By identifying the angles on the unit circle where the x-coordinate is equal to -1/2, we determined the principal solutions: 2π/3 and 4π/3. Recognizing the periodic nature of the cosine function, we expressed the general solution as:
x = 2π/3 + 2πk, where k is an integer x = 4π/3 + 2πk, where k is an integer
Finally, we rigorously verified our solutions by substituting specific values of x from the general solutions back into the original equation, confirming their accuracy and validity.
This exploration underscores the power of combining algebraic techniques with trigonometric principles to solve complex equations. The systematic approach employed in this article serves as a valuable framework for tackling a wide range of trigonometric problems. Moreover, the insights gained into the behavior of trigonometric functions and their applications in various fields highlight the importance of mathematical understanding in the broader context of scientific inquiry.
