Determining The Value Of Cos(A-B) A Trigonometric Exploration
In this comprehensive exploration, we delve into the realm of trigonometry, focusing on the evaluation of trigonometric expressions given specific conditions. Our primary objective is to determine the precise value of cos(A-B), a trigonometric expression that holds significant importance in various mathematical contexts. To embark on this journey, we are provided with crucial information regarding the sine values of angles A and B, along with specific interval restrictions that govern their possible locations within the unit circle. These constraints play a pivotal role in determining the signs of the cosine values associated with these angles, enabling us to arrive at an accurate and reliable solution.
Specifically, we are given that sin(A) = 4/5 and that angle A lies within the interval π/2 < A < π. This signifies that angle A resides in the second quadrant of the unit circle, where sine values are positive and cosine values are negative. Similarly, we are informed that sin(B) = -2√3/5 and that angle B lies within the interval π < B < 3π/2. This places angle B in the third quadrant, where both sine and cosine values are negative. Armed with this information, we are well-equipped to unravel the intricacies of the trigonometric relationship and arrive at the value of cos(A-B).
Before we can directly compute cos(A-B), we must first determine the individual cosine values of angles A and B. This requires us to leverage the fundamental trigonometric identity that connects sine and cosine: sin²(θ) + cos²(θ) = 1. This identity serves as a cornerstone in trigonometry, allowing us to establish a direct relationship between the sine and cosine of any angle. By applying this identity in conjunction with the given sine values and interval restrictions, we can systematically unveil the cosine values of angles A and B.
Let's begin by focusing on angle A. We know that sin(A) = 4/5, so we can substitute this value into the trigonometric identity: (4/5)² + cos²(A) = 1. Simplifying this equation, we get 16/25 + cos²(A) = 1. Subtracting 16/25 from both sides, we have cos²(A) = 9/25. Taking the square root of both sides, we obtain cos(A) = ±3/5. Now, we must consider the interval restriction π/2 < A < π, which indicates that angle A lies in the second quadrant. In the second quadrant, cosine values are negative. Therefore, we confidently conclude that cos(A) = -3/5.
Moving on to angle B, we are given that sin(B) = -2√3/5. Substituting this value into the trigonometric identity, we get (-2√3/5)² + cos²(B) = 1. Simplifying, we have 12/25 + cos²(B) = 1. Subtracting 12/25 from both sides, we obtain cos²(B) = 13/25. Taking the square root of both sides, we get cos(B) = ±√13/5. The interval restriction π < B < 3π/2 tells us that angle B lies in the third quadrant, where cosine values are also negative. Hence, we can definitively say that cos(B) = -√13/5.
With the cosine values of angles A and B now firmly in our grasp, we are ready to tackle the central task of determining the value of cos(A-B). To achieve this, we employ the powerful cosine subtraction formula, a fundamental identity in trigonometry that expresses the cosine of the difference between two angles in terms of their individual sines and cosines. This formula is expressed as follows:
cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
This formula serves as a bridge, connecting the individual trigonometric values of angles A and B to the trigonometric value of their difference. By carefully substituting the values we have already determined, we can systematically evaluate the expression and arrive at the final answer. This formula is a cornerstone of trigonometric manipulation, allowing us to break down complex expressions into simpler components and solve for unknown quantities.
Now, let's put this formula into action. We have established that cos(A) = -3/5, cos(B) = -√13/5, sin(A) = 4/5, and sin(B) = -2√3/5. Substituting these values into the cosine subtraction formula, we get:
cos(A-B) = (-3/5)(-√13/5) + (4/5)(-2√3/5)
Simplifying this expression, we have:
cos(A-B) = (3√13)/25 - (8√3)/25
Combining the terms, we arrive at the final value:
cos(A-B) = (3√13 - 8√3)/25
In this exploration, we have successfully determined the value of cos(A-B), given the sine values of angles A and B and their respective interval restrictions. Our journey began with an understanding of the problem statement, carefully noting the given information and the desired outcome. We then leveraged the fundamental trigonometric identity sin²(θ) + cos²(θ) = 1 to unveil the cosine values of angles A and B, taking into account the sign conventions dictated by their respective quadrants.
With the cosine values in hand, we turned to the cosine subtraction formula, a powerful tool that allows us to express the cosine of the difference between two angles in terms of their individual sines and cosines. By meticulously substituting the values we had determined, we systematically simplified the expression, ultimately arriving at the solution:
cos(A-B) = (3√13 - 8√3)/25
This solution represents the precise value of cos(A-B) under the given conditions. The process we have undertaken highlights the interconnectedness of trigonometric concepts, demonstrating how fundamental identities and formulas can be combined to solve complex problems. It also underscores the importance of considering interval restrictions, which play a crucial role in determining the signs of trigonometric values and ensuring the accuracy of our results.
This exploration serves as a testament to the power of trigonometric principles and their ability to unravel intricate relationships between angles and their associated values. By mastering these concepts, we equip ourselves with the tools to navigate a wide range of mathematical challenges and deepen our understanding of the world around us.
In conclusion, by applying trigonometric identities and considering the given intervals, we found that cos(A-B) = (3√13 - 8√3)/25. This exercise highlights the importance of understanding trigonometric relationships and their applications in solving mathematical problems.
