Solving Sin(θ+30)cos(θ-30) = 0.5 Trigonometric Equation
Introduction
In this comprehensive article, we will delve into the intricacies of solving the trigonometric equation sin(θ+30)cos(θ-30) = 0.5 within the specified range of 0° ≤ θ ≤ 360°. This problem combines trigonometric identities, algebraic manipulation, and a clear understanding of the unit circle. Our goal is to provide a step-by-step solution, making it accessible for students and enthusiasts alike. This exploration includes transforming the equation using trigonometric identities, identifying key angles, and applying general solutions to find all possible values of θ within the given interval. Mastering these techniques is crucial for success in trigonometry and calculus, making this a valuable learning exercise. Throughout this guide, we emphasize clarity and precision to ensure a solid understanding of each step. Understanding trigonometric equations is essential not only for academic success but also for various practical applications in physics, engineering, and other fields. By the end of this article, you will have a robust understanding of how to tackle similar trigonometric problems. We will also highlight common pitfalls and provide tips for avoiding errors, ensuring a smooth problem-solving experience. This detailed approach will empower you to solve a wide range of trigonometric equations with confidence and accuracy.
Problem Statement
Our primary task is to find all values of θ that satisfy the equation sin(θ+30)cos(θ-30) = 0.5 within the interval 0° ≤ θ ≤ 360°. This requires us to use trigonometric identities to simplify the equation and then solve for θ. We will methodically break down the problem, ensuring that each step is clear and understandable. The equation involves trigonometric functions of sums and differences of angles, which necessitates the use of appropriate trigonometric identities. The given range for θ is crucial because it defines the boundaries within which we need to find our solutions. Ignoring this range can lead to incorrect or incomplete solutions. The complexity of the equation lies in the combination of sine and cosine functions with different angle arguments, making it a good exercise for practicing trigonometric manipulations. Successfully solving this problem requires a strong foundation in trigonometric identities and a systematic approach to algebraic manipulation. The problem's structure also allows for a deep dive into understanding how different trigonometric functions interact with each other. Our solution will involve multiple steps, each carefully explained to enhance understanding and retention. This problem not only tests your mathematical skills but also your ability to apply them in a logical and organized manner.
Trigonometric Identities
To solve the equation, we need to employ trigonometric identities to simplify it. Specifically, we will use the product-to-sum identities. The relevant identity here is:
sin(A)cos(B) = 0.5[sin(A+B) + sin(A-B)]
This identity is the cornerstone of our solution. It allows us to convert the product of sine and cosine into a sum of sine functions, which is much easier to handle. Understanding and applying this identity correctly is crucial for simplifying the given equation. The beauty of trigonometric identities lies in their ability to transform complex expressions into simpler forms. This particular identity is widely used in various mathematical and engineering applications, making it an essential tool in the problem-solver's arsenal. Before diving into the application, let's briefly discuss why this identity works. It is derived from the angle sum and difference formulas for sine, which are fundamental trigonometric results. The identity not only simplifies the equation but also provides a pathway to isolate the variable θ. The judicious use of trigonometric identities is a hallmark of elegant problem-solving in mathematics. This identity, in particular, demonstrates the power of transforming products into sums, a technique that is applicable in various contexts. By mastering this identity, you enhance your ability to manipulate trigonometric expressions effectively. This skill is invaluable not only for solving equations but also for proving trigonometric theorems and simplifying complex mathematical models.
Applying the Identity
Using the identity sin(A)cos(B) = 0.5[sin(A+B) + sin(A-B)], let A = θ + 30° and B = θ - 30°. Substituting these values into the identity, we get:
sin(θ + 30°)cos(θ - 30°) = 0.5[sin((θ + 30°) + (θ - 30°)) + sin((θ + 30°) - (θ - 30°))]
Now, let's simplify the equation:
- 5 = 0. 5[sin(2θ) + sin(60°)]
Dividing both sides by 0.5, we obtain:
1 = sin(2θ) + sin(60°)
We know that sin(60°) = √3/2, so the equation becomes:
1 = sin(2θ) + √3/2
This step is a crucial simplification that transforms the original equation into a more manageable form. By correctly applying the trigonometric identity, we have effectively reduced the complexity of the problem. The substitution process requires careful attention to detail to avoid errors. Each term must be correctly placed and simplified. This transformation is not just a mathematical manipulation; it's a strategic move that makes the problem solvable. The simplified equation now involves only a single trigonometric function, sine, of a multiple of θ. This form is much easier to work with, allowing us to isolate and solve for θ. The process highlights the importance of choosing the right identity for the problem at hand. A different identity might not have led to such a clear simplification. The result of this step is a pivotal point in the solution, paving the way for the final answer. By mastering this technique, you gain the ability to transform complex trigonometric expressions into simpler, solvable forms.
Isolating sin(2θ)
To isolate sin(2θ), subtract √3/2 from both sides of the equation:
sin(2θ) = 1 - √3/2
Now we have sin(2θ) equal to a specific value. This step is essential for solving the equation because it isolates the trigonometric function. The value 1 - √3/2 is a constant, and we need to find the angles for which the sine function equals this value. This isolation step is a common technique in solving trigonometric equations. It allows us to focus on the trigonometric function and its argument, making it easier to find solutions. The subtraction operation is straightforward, but it's crucial to perform it accurately to maintain the integrity of the equation. The resulting expression, sin(2θ) = 1 - √3/2, is in a standard form for solving trigonometric equations. This form allows us to use the inverse sine function to find the reference angle and then determine the general solutions. The process of isolating the trigonometric function is not just a mathematical step; it’s a strategic move that brings us closer to the solution. The isolated form highlights the relationship between the angle 2θ and its sine value, making it easier to identify potential solutions within the given range.
Finding the Reference Angle
Let α be the reference angle such that:
sin(α) = 1 - √3/2
Using a calculator, we find the approximate value of α:
α ≈ sin⁻¹(1 - √3/2) ≈ 7.60°
This reference angle is crucial for finding all solutions within the given range. The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. It helps us to identify all angles with the same sine value in different quadrants. The inverse sine function, sin⁻¹, gives us the principal value, which is the angle in the range -90° to 90°. However, sine is also positive in the second quadrant, so we need to consider angles in that quadrant as well. This step requires a good understanding of the unit circle and the properties of trigonometric functions in different quadrants. The approximation of α using a calculator is a practical step, but it's essential to remember that the exact value might be necessary for some applications. The reference angle acts as a building block for constructing the general solutions. It provides a starting point for identifying all possible angles that satisfy the equation. Finding the reference angle is a pivotal step in solving trigonometric equations, as it allows us to systematically determine all solutions within the specified interval. This process not only enhances your problem-solving skills but also deepens your understanding of trigonometric functions and their properties.
General Solutions for 2θ
Since sin(2θ) = 1 - √3/2, we need to find the general solutions for 2θ. Sine is positive in the first and second quadrants. Therefore, the general solutions are:
2θ = α + 360°k or 2θ = (180° - α) + 360°k, where k is an integer.
Substituting the approximate value of α ≈ 7.60°:
2θ ≈ 7.60° + 360°k or 2θ ≈ (180° - 7.60°) + 360°k
2θ ≈ 7.60° + 360°k or 2θ ≈ 172.40° + 360°k
These general solutions represent all possible angles 2θ that satisfy the equation. The general solutions are expressed in terms of the reference angle and integer multiples of 360° because trigonometric functions are periodic with a period of 360°. Considering both the first and second quadrants is crucial for finding all possible solutions. The integer k allows us to account for all coterminal angles, which are angles that have the same terminal side. This step requires a strong understanding of the periodicity of trigonometric functions and the concept of general solutions. The general solutions provide a comprehensive set of angles that satisfy the equation, but we need to consider the specific range 0° ≤ θ ≤ 360° to find the relevant solutions. This step is a bridge between the general solutions and the specific solutions within the given interval. The ability to express solutions in general form is a fundamental skill in trigonometry, and it's essential for solving more complex problems. The general solutions not only provide a complete set of angles but also highlight the periodic nature of trigonometric functions, a concept that is crucial in many areas of mathematics and physics.
Solving for θ
Now, divide both sides of the general solutions by 2 to solve for θ:
θ ≈ 3.80° + 180°k or θ ≈ 86.20° + 180°k
This gives us the general solutions for θ. The division by 2 is a straightforward algebraic step, but it's crucial for isolating θ. These general solutions for θ represent all possible angles that satisfy the original equation within the specified range. Each general solution consists of a particular angle plus integer multiples of 180°, reflecting the periodicity of the solutions. Now, we need to find the specific values of θ within the range 0° ≤ θ ≤ 360° by substituting different integer values for k. This step is a critical part of the solution process, as it narrows down the infinite set of general solutions to the relevant solutions within the given interval. The division step is not just a mathematical operation; it's a key step in unraveling the argument of the sine function and obtaining the desired angle θ. The general solutions for θ provide a framework for identifying all potential solutions, and the next step involves selecting the solutions that fall within the specified range. The ability to manipulate and solve for variables in general solutions is a valuable skill in mathematics, applicable to various types of equations.
Finding Specific Solutions in the Range 0° ≤ θ ≤ 360°
We need to find the values of θ within the interval 0° ≤ θ ≤ 360°. Let's substitute different integer values for k in both general solutions:
For θ ≈ 3.80° + 180°k:
- k = 0: θ ≈ 3.80°
- k = 1: θ ≈ 183.80°
- k = 2: θ ≈ 363.80° (This is outside the range, so we discard it)
For θ ≈ 86.20° + 180°k:
- k = 0: θ ≈ 86.20°
- k = 1: θ ≈ 266.20°
- k = 2: θ ≈ 446.20° (This is outside the range, so we discard it)
Thus, the solutions within the range 0° ≤ θ ≤ 360° are approximately:
θ ≈ 3.80°, 86.20°, 183.80°, 266.20°
This step is the culmination of our solution process. It involves systematically substituting integer values for k to identify the solutions that fall within the specified range. The process of checking each solution against the range is crucial to ensure accuracy. Solutions outside the range are discarded, leaving us with the valid solutions. This step requires careful attention to detail and a systematic approach to avoid missing any solutions. The substitution process highlights the practical application of the general solutions. It demonstrates how general solutions can be used to find specific solutions in a given context. The identified solutions are the angles that satisfy the original equation within the specified interval. This step not only provides the final answers but also reinforces the understanding of how general solutions and ranges interact. The ability to find specific solutions from general solutions is a valuable skill in mathematics, applicable to various problem-solving scenarios.
Final Answer
The solutions to the equation sin(θ+30)cos(θ-30) = 0.5 for 0° ≤ θ ≤ 360° are approximately:
θ ≈ 3.80°, 86.20°, 183.80°, 266.20°
These values represent the angles within the specified range that satisfy the given trigonometric equation. The final answer is the result of a methodical and detailed solution process. It encapsulates the application of trigonometric identities, algebraic manipulation, and an understanding of general solutions. Each solution has been carefully verified against the given range to ensure accuracy. The final answer not only provides the numerical values but also represents the successful application of problem-solving techniques. It demonstrates the power of combining different mathematical concepts to solve a complex problem. The final answer is a testament to the importance of precision and clarity in mathematical reasoning. It is the result of a step-by-step approach, where each step builds upon the previous one to arrive at the solution. The final answer is not just a set of numbers; it’s a culmination of the entire problem-solving journey. It reinforces the understanding of trigonometric equations and their solutions. The ability to arrive at a correct final answer with confidence is a hallmark of mathematical proficiency.
Conclusion
In this article, we have provided a detailed solution to the trigonometric equation sin(θ+30)cos(θ-30) = 0.5 for 0° ≤ θ ≤ 360°. We started by simplifying the equation using trigonometric identities, specifically the product-to-sum identity. Then, we isolated the sine function, found the reference angle, and determined the general solutions. Finally, we identified the specific solutions within the given range. This process demonstrates the importance of a systematic approach to problem-solving in mathematics. The successful solution of this equation requires a strong foundation in trigonometric identities, algebraic manipulation, and an understanding of the unit circle. The steps involved highlight the interconnectedness of various mathematical concepts. Solving trigonometric equations is not just about finding the correct answer; it's about developing a deep understanding of the underlying principles. The skills learned in this process are applicable to a wide range of mathematical and scientific problems. By mastering these techniques, you can confidently tackle complex trigonometric equations and related challenges. The detailed explanation provided in this article aims to empower you with the knowledge and skills necessary to succeed in trigonometry and beyond. The process of solving this equation serves as a valuable learning experience, reinforcing your understanding of trigonometric concepts and problem-solving strategies.
