Solving $2 \cos ^3 X-5 \cos ^2 X+\cos X+2=0$ A Step-by-Step Guide

Introduction: Unraveling Trigonometric Mysteries

Trigonometric equations, at first glance, may appear daunting with their mix of cosine, sine, tangent, and other functions. However, many of these equations can be solved using algebraic techniques combined with trigonometric identities. In this article, we will delve into the process of solving the specific trigonometric equation 2 cos³ x - 5 cos² x + cos x + 2 = 0. We will explore the steps involved in transforming the equation into a more manageable form, identifying its roots, and ultimately finding the general solutions. Understanding how to tackle such equations is crucial for anyone studying mathematics, physics, engineering, or any field where periodic phenomena are modeled.

Before we dive into the specifics, it's important to recall some fundamental concepts. The cosine function, denoted as cos(x), is a periodic function that oscillates between -1 and 1. Its values repeat every 2π radians or 360 degrees. This periodicity is key to understanding the multiple solutions that trigonometric equations can have. When solving for x, we are essentially looking for the angles that satisfy the given equation. Due to the periodic nature of trigonometric functions, there will be infinitely many such angles, which can be expressed in a general form.

This article aims to provide a comprehensive guide to solving the given trigonometric equation, breaking down each step into manageable parts. By the end of this discussion, you will have a clear understanding of the techniques involved and be able to apply them to similar problems. So, let's embark on this mathematical journey and unlock the secrets hidden within this trigonometric equation.

Transforming the Equation: A Polynomial Approach

Our initial equation is 2 cos³ x - 5 cos² x + cos x + 2 = 0. The first step in solving this equation is to recognize its structure. Notice that if we substitute y = cos x, the equation transforms into a cubic polynomial equation: 2y³ - 5y² + y + 2 = 0. This transformation is a critical step because we can now leverage our knowledge of polynomial algebra to find the roots of the equation. By treating cos x as a variable, we can simplify the problem and apply techniques such as factoring or the rational root theorem.

The rational root theorem states that if a polynomial equation with integer coefficients has a rational root p/q (where p and q are integers with no common factors other than 1), then p must be a factor of the constant term and q must be a factor of the leading coefficient. In our case, the constant term is 2 and the leading coefficient is 2. Therefore, the possible rational roots are ±1, ±2, and ±1/2. We can test these possible roots by substituting them into the polynomial equation and checking if the equation holds true.

Let's start by testing y = 1. Substituting y = 1 into the polynomial equation gives us 2(1)³ - 5(1)² + 1 + 2 = 2 - 5 + 1 + 2 = 0. This confirms that y = 1 is indeed a root of the polynomial equation. Knowing one root allows us to factor the polynomial. We can use synthetic division or polynomial long division to divide the polynomial by (y - 1). Performing synthetic division, we find that 2y³ - 5y² + y + 2 divided by (y - 1) gives us 2y² - 3y - 2. Therefore, we can rewrite the polynomial equation as (y - 1)(2y² - 3y - 2) = 0. This factorization is a significant step forward, as it breaks down the cubic polynomial into a linear factor and a quadratic factor, which are easier to solve.

Now we need to solve the quadratic equation 2y² - 3y - 2 = 0. We can use the quadratic formula, factoring, or completing the square to find the roots. In this case, the quadratic equation can be factored as (2y + 1)(y - 2) = 0. This gives us two more roots: y = -1/2 and y = 2. Thus, the roots of the polynomial equation 2y³ - 5y² + y + 2 = 0 are y = 1, y = -1/2, and y = 2. These roots are crucial for finding the solutions to our original trigonometric equation, as they represent the possible values of cos x.

Finding the Solutions for x: Back to Trigonometry

Having found the roots of the polynomial equation, we now need to relate these roots back to the original trigonometric equation. Recall that we made the substitution y = cos x. Therefore, the roots y = 1, y = -1/2, and y = 2 correspond to the equations cos x = 1, cos x = -1/2, and cos x = 2. It's crucial to remember that the range of the cosine function is [-1, 1]. This means that the value of cos x can never be 2. Therefore, the equation cos x = 2 has no solutions, and we can disregard this case. We are left with two equations to solve: cos x = 1 and cos x = -1/2.

Let's first consider the equation cos x = 1. The cosine function equals 1 at angles that are multiples of 2π radians (or 360 degrees). This is because the cosine function represents the x-coordinate of a point on the unit circle, and this coordinate is 1 at 0 radians, 2π radians, 4π radians, and so on. Therefore, the general solution for cos x = 1 is x = 2nπ, where n is an integer. This represents all angles that are coterminal with 0 radians.

Next, we need to solve the equation cos x = -1/2. This equation is slightly more complex as the cosine function equals -1/2 at two different angles within the interval [0, 2π). To find these angles, we can use the unit circle or the inverse cosine function. The reference angle for cos x = 1/2 is π/3 radians (or 60 degrees). Since cosine is negative in the second and third quadrants, the angles where cos x = -1/2 are x = π - π/3 = 2π/3 radians and x = π + π/3 = 4π/3 radians. These are the two solutions within one period of the cosine function.

To find the general solutions for cos x = -1/2, we need to add multiples of 2π to these specific solutions. This gives us the general solutions x = 2π/3 + 2nπ and x = 4π/3 + 2nπ, where n is an integer. These expressions represent all angles that have a cosine of -1/2.

In summary, the solutions to the original trigonometric equation 2 cos³ x - 5 cos² x + cos x + 2 = 0 are x = 2nπ, x = 2π/3 + 2nπ, and x = 4π/3 + 2nπ, where n is an integer. These solutions represent all angles that satisfy the equation. By breaking down the problem into smaller parts, using algebraic techniques, and understanding the properties of trigonometric functions, we were able to find the complete solution set.

General Solutions and Their Significance

The general solutions we've found, x = 2nπ, x = 2π/3 + 2nπ, and x = 4π/3 + 2nπ, where n is an integer, represent an infinite set of angles that satisfy the equation 2 cos³ x - 5 cos² x + cos x + 2 = 0. The integer 'n' allows us to capture all possible solutions by adding multiples of 2π to the principal solutions. Understanding the concept of general solutions is crucial in trigonometry, as it highlights the periodic nature of trigonometric functions and the multiple angles that can yield the same function value.

The solution x = 2nπ represents all angles coterminal with 0 radians, where the cosine function equals 1. This is a fundamental solution that arises frequently in trigonometric equations. The solutions x = 2π/3 + 2nπ and x = 4π/3 + 2nπ are more specific, representing angles where the cosine function equals -1/2. These solutions demonstrate how trigonometric equations can have multiple solutions within a given interval, and the general solutions capture all of them.

It's important to note that when solving trigonometric equations, it's often necessary to provide general solutions unless the problem specifies a particular interval. For instance, if the problem asked for solutions in the interval [0, 2π), we would only consider the solutions within that range, which would be 0, 2π/3, and 4π/3. However, the general solutions provide a complete picture of all possible angles that satisfy the equation.

In practical applications, understanding general solutions is essential. For example, in physics, when modeling oscillations or wave phenomena, the general solutions of trigonometric equations can describe the motion or behavior of a system over time. The integer 'n' can represent the number of cycles or periods that have occurred, allowing us to predict the state of the system at any given time. Similarly, in engineering, general solutions are used in the design of systems that involve periodic signals or vibrations.

Therefore, while finding the principal solutions within a specific interval is important, grasping the concept of general solutions is crucial for a comprehensive understanding of trigonometry and its applications. It allows us to move beyond specific cases and appreciate the broader picture of trigonometric functions and their behavior.

Conclusion: Mastering Trigonometric Equations

In this article, we have successfully navigated the complexities of solving the trigonometric equation 2 cos³ x - 5 cos² x + cos x + 2 = 0. We began by transforming the equation into a polynomial form using the substitution y = cos x. This allowed us to leverage our knowledge of polynomial algebra to find the roots of the equation. We then related these roots back to the original trigonometric equation, solving for x and finding the general solutions. This process involved a combination of algebraic manipulation, trigonometric identities, and a deep understanding of the properties of the cosine function.

The key takeaways from this discussion include:

• Transformation: Recognizing that trigonometric equations can often be transformed into algebraic equations is a crucial step in solving them.
• Factoring: Factoring polynomials is a powerful technique for finding roots and simplifying equations.
• Rational Root Theorem: The rational root theorem can help identify potential rational roots of polynomial equations.
• General Solutions: Understanding the periodic nature of trigonometric functions and expressing solutions in general form is essential.
• Unit Circle: The unit circle is a valuable tool for visualizing trigonometric functions and finding solutions to equations.

By mastering these techniques, you can confidently tackle a wide range of trigonometric equations. Remember that practice is key. The more you solve such equations, the more comfortable you will become with the process. Don't be afraid to break down complex problems into smaller, more manageable steps. And always keep in mind the fundamental properties of trigonometric functions, such as their periodicity and range.

Trigonometric equations are not just abstract mathematical exercises. They have real-world applications in various fields, including physics, engineering, computer science, and more. Understanding how to solve these equations is a valuable skill that can open doors to a deeper understanding of the world around us. So, continue to explore the fascinating world of trigonometry and embrace the challenges it presents. With perseverance and the right tools, you can unlock the secrets hidden within these equations and gain a profound appreciation for the beauty and power of mathematics.