Solve The Equation 2sin^2x - 3sinxcosx - 5cos^2x = 0.
Introduction
This article delves into the solution of the trigonometric equation 2sin^2x - 3sinxcosx - 5cos^2x = 0. Trigonometric equations like this one appear frequently in various fields of mathematics, physics, and engineering. Understanding how to solve them is a crucial skill. We will explore a step-by-step approach to find the general solutions for x that satisfy the given equation. This process will involve algebraic manipulation, trigonometric identities, and a keen understanding of the periodic nature of trigonometric functions.
Step 1: Recognizing the Homogeneous Nature
In tackling the trigonometric equation 2sin^2x - 3sinxcosx - 5cos^2x = 0, the first key step involves recognizing its homogeneous nature. This equation is homogeneous because every term has the same total degree (which is 2 in this case). Specifically, sin^2x has a degree of 2, sinxcosx has a degree of 1 + 1 = 2, and cos^2x also has a degree of 2. This homogeneity is crucial because it allows us to simplify the equation by dividing through by a suitable trigonometric term. Recognizing this characteristic is a fundamental skill in solving trigonometric equations, as it guides us toward a more manageable form. When dealing with equations where all terms have the same degree in trigonometric functions, this approach is often a highly effective starting point. By identifying and leveraging this homogeneity, we can transform a seemingly complex equation into a simpler one that is more amenable to solution.
The importance of recognizing the homogeneous nature of the equation cannot be overstated. It sets the stage for the subsequent steps in the solution process and significantly simplifies the overall complexity. Without this recognition, we might attempt other less efficient or even incorrect methods, leading to unnecessary complications or incorrect answers. Therefore, developing an eye for spotting homogeneity is an invaluable asset in the realm of trigonometric problem-solving. This recognition paves the way for the next logical step, which involves dividing the equation by a trigonometric term. This division effectively reduces the equation into a form that is expressible in terms of a single trigonometric function, thereby streamlining the solution process and making it more accessible. It’s a strategic move that transforms the landscape of the problem, making it much easier to navigate toward a solution.
Furthermore, understanding homogeneity extends beyond this specific equation. Many trigonometric problems can be simplified by recognizing and exploiting this property. It's a powerful concept that provides a valuable tool for solving a broad spectrum of equations. This technique isn't limited to second-degree equations; it can be applied to equations of higher degrees as well, provided they maintain the homogeneous property. The principle remains consistent: identifying terms with the same degree allows for strategic simplification, making the equation more tractable. This skill is not only applicable in purely mathematical contexts but also in various applied fields where trigonometric equations arise, such as physics, engineering, and computer graphics. Homogeneity, in essence, is a fundamental concept that provides a pathway to simplify and solve a diverse range of problems, highlighting its importance in the toolkit of any problem solver.
Step 2: Dividing by cos^2x
Following the identification of the equation's homogeneous nature, the next critical step in solving 2sin^2x - 3sinxcosx - 5cos^2x = 0 involves dividing both sides of the equation by cos^2x. This is a strategic move predicated on the fact that we can express the equation in terms of the tangent function (tan x) after this division. However, before proceeding with the division, it's crucial to consider the potential for cos x to be zero. If cos x = 0, then x would be of the form (2n + 1)π/2, where n is an integer. We need to check if these values satisfy the original equation.
If cos x = 0, then sin x would be ±1. Substituting these values into the original equation, we get: 2(±1)^2 - 3(±1)(0) - 5(0)^2 = 2, which is not equal to 0. Therefore, cos x = 0 is not a solution, and we are safe to divide by cos^2x without losing any potential solutions. This step of verifying the potential for zero-division is a cornerstone of rigorous mathematical problem-solving, particularly in trigonometric equations, where the periodic nature of functions can introduce such scenarios. Failing to check this condition could lead to the omission of solutions or the inclusion of extraneous ones, undermining the accuracy of the final result. Thus, this careful consideration exemplifies the methodical approach required for solving trigonometric problems effectively.
Now, dividing each term of the equation by cos^2x, we obtain a new equation expressed in terms of tan x. This transformation is significant because it reduces the equation's complexity, making it easier to solve. Specifically, the terms sin2x/cos2x become tan^2x, (sinxcosx)/cos^2x becomes tan x, and cos2x/cos2x becomes 1. This process effectively converts the trigonometric equation into an algebraic one, specifically a quadratic equation in tan x. This transition is a key technique in solving trigonometric equations, as it allows us to leverage algebraic methods to find solutions. The resulting quadratic equation is typically much simpler to solve than the original trigonometric form, marking a pivotal step in the overall solution strategy. This highlights the power of algebraic manipulation in simplifying complex trigonometric expressions, providing a clear pathway toward finding the solutions.
Step 3: Forming a Quadratic Equation in tan x
After dividing the original equation 2sin^2x - 3sinxcosx - 5cos^2x = 0 by cos^2x, we arrive at a new equation: 2tan^2x - 3tanx - 5 = 0. This equation is a quadratic equation in terms of tan x. Recognizing this is a crucial step because it allows us to apply familiar algebraic techniques to solve for tan x. The transformation from trigonometric form to this algebraic form is a powerful method in solving trigonometric equations. By identifying the quadratic structure, we can employ standard methods like factoring, completing the square, or using the quadratic formula to find the values of tan x that satisfy the equation.
The general form of a quadratic equation is ax^2 + bx + c = 0, and our equation 2tan^2x - 3tanx - 5 = 0 perfectly fits this form, where 'x' is replaced by 'tan x', a = 2, b = -3, and c = -5. This alignment with the standard quadratic form allows us to directly apply established algebraic solution techniques. The strategic conversion of the trigonometric equation into this quadratic form is a testament to the power of mathematical manipulation. It transforms a potentially daunting trigonometric problem into a more manageable algebraic one. This step is not just a mathematical maneuver but a strategic simplification that paves the way for a straightforward solution. The ability to recognize such structures and apply appropriate methods is a hallmark of proficiency in mathematical problem-solving. It showcases the interconnectedness of different mathematical concepts and the power of using algebraic tools within a trigonometric context.
Once the equation is recognized as a quadratic in tan x, the next logical step is to solve it using methods applicable to quadratic equations. This often involves factoring, which, if possible, is a direct and efficient method. Alternatively, the quadratic formula provides a guaranteed solution, regardless of whether the equation can be easily factored. The goal at this stage is to isolate tan x, finding the specific numerical values that satisfy the quadratic equation. These values of tan x will then be used in subsequent steps to determine the values of x that solve the original trigonometric equation. Therefore, the process of forming and recognizing the quadratic equation in tan x is a critical juncture in the overall solution strategy, bridging trigonometric expressions with algebraic techniques and setting the stage for the final determination of the solution set.
Step 4: Solving the Quadratic Equation
Now that we have the quadratic equation 2tan^2x - 3tanx - 5 = 0, we need to solve it for tan x. This can be achieved through factoring, the quadratic formula, or completing the square. In this case, factoring is a straightforward approach. We look for two numbers that multiply to (2)(-5) = -10 and add up to -3. These numbers are -5 and 2. Therefore, we can rewrite the middle term as -5tan x + 2tan x.
This gives us: 2tan^2x - 5tanx + 2tanx - 5 = 0. Now, we can factor by grouping: tanx(2tanx - 5) + 1(2tanx - 5) = 0. This leads to (tanx + 1)(2tanx - 5) = 0. Setting each factor equal to zero gives us two possible solutions for tan x: tanx + 1 = 0, which implies tan x = -1, and 2tanx - 5 = 0, which implies tan x = 5/2. This factorization is a crucial step, as it decomposes the quadratic equation into two simpler linear equations, each of which can be easily solved for tan x. The skill of factoring quadratic expressions is fundamental in algebra and is frequently applied in various mathematical contexts, including solving trigonometric equations. The ability to efficiently factor quadratic equations like this one can significantly streamline the solution process.
Alternatively, if factoring is not immediately apparent, the quadratic formula can be used. For a quadratic equation of the form ax^2 + bx + c = 0, the quadratic formula is given by x = [-b ± √(b^2 - 4ac)] / (2a). Applying this to our equation 2tan^2x - 3tanx - 5 = 0, where a = 2, b = -3, and c = -5, we get: tan x = [3 ± √((-3)^2 - 4(2)(-5))] / (2(2)). Simplifying this, we get: tan x = [3 ± √(9 + 40)] / 4 = [3 ± √49] / 4 = [3 ± 7] / 4. This gives us two solutions: tan x = (3 + 7) / 4 = 10 / 4 = 5/2 and tan x = (3 - 7) / 4 = -4 / 4 = -1. As we can see, the quadratic formula provides the same solutions as factoring, demonstrating its versatility and reliability as a method for solving quadratic equations. Regardless of the method used, solving for tan x is a pivotal step in finding the solutions to the original trigonometric equation.
Step 5: Finding the General Solutions for x
Having found the values of tan x, we now need to find the general solutions for x. We have two cases to consider: tan x = -1 and tan x = 5/2. For tan x = -1, we know that the tangent function is -1 at angles 3π/4 and 7π/4 in the interval [0, 2π). Since the tangent function has a period of π, the general solution for tan x = -1 is given by: x = 3π/4 + nπ, where n is an integer. This general solution captures all angles that have a tangent of -1, accounting for the periodic nature of the tangent function. The 'nπ' term represents the infinite repetitions of the solution that occur every π radians, reflecting the cyclical pattern inherent in trigonometric functions. This understanding of periodicity is essential when solving trigonometric equations, as it allows us to express the complete set of solutions.
For tan x = 5/2, we need to use the arctangent function (tan^-1) to find the principal value of x. Let's call the principal value α, so α = tan^-1(5/2). Using a calculator, we find that α ≈ 1.190 radians (approximately 68.199 degrees). Again, considering the periodicity of the tangent function, the general solution for tan x = 5/2 is given by: x = α + nπ, where n is an integer. This general solution encompasses all angles that have a tangent of 5/2, extending the principal value α by integer multiples of π. The use of the arctangent function is a standard technique in solving trigonometric equations when the values are not standard angles (like 0, π/6, π/4, etc.). It provides a way to find the base angle, which is then used to generate the complete set of solutions by adding multiples of the period.
Combining both cases, the general solutions for the original equation 2sin^2x - 3sinxcosx - 5cos^2x = 0 are: x = 3π/4 + nπ and x = tan^-1(5/2) + nπ, where n is an integer. These solutions represent all possible values of x that satisfy the equation. It’s important to express the solutions in general form to account for the periodic nature of trigonometric functions. These general solutions capture the infinite set of angles that make the equation true, highlighting the comprehensive nature of the solution process. Therefore, the final step of finding the general solutions is crucial in providing a complete and accurate answer to the trigonometric equation.
Conclusion
In conclusion, solving the trigonometric equation 2sin^2x - 3sinxcosx - 5cos^2x = 0 involves several key steps: recognizing the homogeneous nature of the equation, dividing by cos^2x to form a quadratic equation in tan x, solving the quadratic equation, and finally, finding the general solutions for x. The general solutions are x = 3π/4 + nπ and x = tan^-1(5/2) + nπ, where n is an integer. This process demonstrates a systematic approach to solving trigonometric equations, emphasizing the importance of algebraic manipulation, trigonometric identities, and understanding the periodic nature of trigonometric functions. By following these steps, one can effectively solve a wide range of trigonometric equations.