Finding Sin(A+B) Trigonometric Identity Explained

In the fascinating world of trigonometry, we often encounter problems that require us to manipulate and simplify trigonometric expressions. One such challenge involves finding the sine of the sum of two angles, sin(A+B), when given information about the individual angles A and B. This article delves into a classic trigonometry problem, exploring the concepts, steps, and reasoning behind finding sin(A+B). We'll break down the problem, providing a clear and concise solution, while also highlighting the key trigonometric identities and principles involved. By the end of this exploration, you'll have a solid understanding of how to tackle similar trigonometric challenges with confidence. Let's embark on this journey of mathematical discovery and unravel the secrets of trigonometric identities, focusing specifically on how to determine sin(A+B) when provided with the sine of angle A and the cosine of angle B, along with their respective quadrants. This knowledge is not only crucial for solving mathematical problems but also for understanding the broader applications of trigonometry in various fields such as physics, engineering, and computer graphics.

Let's consider a scenario where we are given that sin(A) = 8/17, with angle A residing in Quadrant II, and cos(B) = 4/5, with angle B in Quadrant I. Our mission is to find the value of sin(A+B) and express our answer as a fraction. This problem encapsulates several fundamental concepts in trigonometry, including the definitions of trigonometric functions in different quadrants, the Pythagorean identity, and the sine addition formula. To successfully navigate this problem, we must first understand the implications of the given information. The quadrant in which an angle lies dictates the signs of its trigonometric functions. In Quadrant II, sine is positive, while cosine and tangent are negative. Conversely, in Quadrant I, all trigonometric functions are positive. By carefully considering these quadrant restrictions, we can accurately determine the signs of the trigonometric values we need. Furthermore, the Pythagorean identity, sin²(θ) + cos²(θ) = 1, is a cornerstone of trigonometry and allows us to find the missing trigonometric function value for a given angle. In this case, we can use the Pythagorean identity to find cos(A) from sin(A) and sin(B) from cos(B). Finally, the sine addition formula, sin(A+B) = sin(A)cos(B) + cos(A)sin(B), provides the key to combining the individual trigonometric values to find sin(A+B). By mastering these core concepts and applying them systematically, we can confidently solve this problem and gain a deeper appreciation for the beauty and power of trigonometric identities.

Step 1: Finding cos(A)

We are given that sin(A) = 8/17 and that A is in Quadrant II. In Quadrant II, cosine is negative. To find cos(A), we can use the Pythagorean identity: sin²(A) + cos²(A) = 1. Substituting the given value of sin(A), we have (8/17)² + cos²(A) = 1. This simplifies to 64/289 + cos²(A) = 1. Subtracting 64/289 from both sides, we get cos²(A) = 1 - 64/289 = 225/289. Taking the square root of both sides, we find cos(A) = ±√(225/289) = ±15/17. Since A is in Quadrant II, cos(A) must be negative. Therefore, cos(A) = -15/17. This step is crucial because it provides us with the cosine value of angle A, which is essential for applying the sine addition formula. The Pythagorean identity serves as a fundamental tool in trigonometry, allowing us to relate sine and cosine values for the same angle. By carefully considering the quadrant in which the angle lies, we can determine the correct sign for the trigonometric function. In this case, the negative sign for cos(A) reflects the fact that cosine is negative in Quadrant II. This careful attention to detail ensures the accuracy of our calculations and the correctness of our final answer.

Step 2: Finding sin(B)

We are given that cos(B) = 4/5 and that B is in Quadrant I. In Quadrant I, sine is positive. To find sin(B), we again use the Pythagorean identity: sin²(B) + cos²(B) = 1. Substituting the given value of cos(B), we have sin²(B) + (4/5)² = 1. This simplifies to sin²(B) + 16/25 = 1. Subtracting 16/25 from both sides, we get sin²(B) = 1 - 16/25 = 9/25. Taking the square root of both sides, we find sin(B) = ±√(9/25) = ±3/5. Since B is in Quadrant I, sin(B) must be positive. Therefore, sin(B) = 3/5. Similar to the previous step, this step utilizes the Pythagorean identity to find the sine value of angle B, given its cosine value. The fact that angle B lies in Quadrant I dictates that sin(B) is positive. This step reinforces the importance of considering the quadrant when determining the signs of trigonometric functions. The value of sin(B) is now readily available for use in the sine addition formula, bringing us closer to our final solution. The consistent application of the Pythagorean identity and careful consideration of quadrant restrictions are hallmarks of successful problem-solving in trigonometry.

Step 3: Applying the Sine Addition Formula

Now that we have found cos(A) = -15/17 and sin(B) = 3/5, we can use the sine addition formula: sin(A+B) = sin(A)cos(B) + cos(A)sin(B). Substituting the given and calculated values, we get sin(A+B) = (8/17)(4/5) + (-15/17)(3/5). This simplifies to sin(A+B) = 32/85 - 45/85. Combining the fractions, we have sin(A+B) = (32 - 45)/85 = -13/85. Therefore, sin(A+B) = -13/85. This step is the culmination of our efforts, where we apply the sine addition formula to combine the individual trigonometric values and arrive at the final answer. The formula itself is a fundamental trigonometric identity that expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles. The substitution of the previously calculated values of sin(A), cos(A), sin(B), and cos(B) is a straightforward process, but it is crucial to maintain accuracy in the arithmetic operations. The resulting fraction, -13/85, represents the value of sin(A+B) for the given conditions. This final answer demonstrates the power of trigonometric identities in simplifying complex expressions and solving trigonometric problems. The negative sign of the result indicates that the angle (A+B) lies in a quadrant where sine is negative, which is consistent with the given information about angles A and B.

Therefore, sin(A+B) = -13/85

In this article, we successfully navigated a trigonometric problem involving finding sin(A+B) given information about sin(A), cos(B), and their respective quadrants. We began by understanding the problem statement and the key trigonometric concepts involved, including quadrant restrictions, the Pythagorean identity, and the sine addition formula. We then systematically worked through the solution, first finding cos(A) using the Pythagorean identity and the quadrant restriction, then finding sin(B) using the same approach. Finally, we applied the sine addition formula to combine these values and arrive at the final answer, sin(A+B) = -13/85. This problem-solving process highlights the importance of a structured approach in mathematics. By breaking down a complex problem into smaller, manageable steps, we can tackle it with greater confidence and accuracy. Furthermore, this exercise reinforces the fundamental concepts of trigonometry, such as the Pythagorean identity and the sine addition formula, which are essential tools in any mathematical toolbox. Understanding the relationship between trigonometric functions and their quadrants is also crucial for accurate problem-solving. The ability to manipulate trigonometric expressions and solve trigonometric equations is not only valuable in mathematics but also in various fields such as physics, engineering, and computer science. By mastering these skills, we can unlock a deeper understanding of the world around us and apply mathematical principles to solve real-world problems. In conclusion, this exploration of finding sin(A+B) serves as a testament to the power and elegance of trigonometry and its ability to connect seemingly disparate concepts into a cohesive framework.