Finding Sin(a+b) Given Sin(a) And Cos(b) In The Fourth Quadrant

In trigonometry, finding the sine of the sum of two angles, sin(a + b), is a common problem. This article provides a detailed solution to finding sin(a + b) when given sin(a) and cos(b), with both angles a and b lying in the interval (3π/2, 2π). This interval places both angles in the fourth quadrant, where sine is negative and cosine is positive. We will explore the steps to solve this problem, ensuring a clear and comprehensive understanding.

Our main objective is to find the value of sin(a + b), given that sin(a) = -2/5 and cos(b) = 1/3, with both a and b in the interval (3π/2, 2π). This places both angles in the fourth quadrant. To find sin(a + b), we use the trigonometric identity:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

We are given sin(a) and cos(b), but we need to find cos(a) and sin(b). Since a and b are in the fourth quadrant, we know that cos(a) is positive and sin(b) is negative. This knowledge is crucial for determining the correct signs of the values we calculate. Understanding the quadrants and their corresponding trigonometric signs is fundamental in solving trigonometric problems. The fourth quadrant is unique in that it is where cosine attains positive values while sine remains negative. This characteristic will greatly influence the calculations and final result, ensuring accuracy and correctness in our trigonometric evaluations.

To find cos(a), we use the Pythagorean identity:

sin²(a) + cos²(a) = 1

Substituting the given value of sin(a) = -2/5, we get:

(-2/5)² + cos²(a) = 1

4/25 + cos²(a) = 1

cos²(a) = 1 - 4/25

cos²(a) = 21/25

Taking the square root of both sides:

cos(a) = ±√(21/25)

cos(a) = ±√21 / 5

Since a is in the fourth quadrant, cos(a) is positive. Therefore:

cos(a) = √21 / 5

This calculation exemplifies the importance of the Pythagorean identity in linking sine and cosine functions. The identity is a cornerstone in trigonometry, allowing us to derive one trigonometric function's value from another. The correct identification of the sign of cos(a) as positive in the fourth quadrant showcases a deep understanding of trigonometric functions' behavior across different quadrants. This step is pivotal in ensuring that the subsequent calculations and the final solution remain accurate. The methodical approach taken here underlines the necessity of clear, step-by-step problem-solving in trigonometry.

Similarly, to find sin(b), we use the Pythagorean identity:

sin²(b) + cos²(b) = 1

Substituting the given value of cos(b) = 1/3, we get:

sin²(b) + (1/3)² = 1

sin²(b) + 1/9 = 1

sin²(b) = 1 - 1/9

sin²(b) = 8/9

Taking the square root of both sides:

sin(b) = ±√(8/9)

sin(b) = ±2√2 / 3

Since b is in the fourth quadrant, sin(b) is negative. Therefore:

sin(b) = -2√2 / 3

The calculation of sin(b) mirrors the process used for finding cos(a), highlighting the consistent application of the Pythagorean identity. This step further emphasizes the critical role of understanding the sign conventions in different quadrants. The determination of sin(b) as negative in the fourth quadrant is a key aspect of maintaining precision and accuracy throughout the problem-solving process. This meticulous attention to detail is what ensures the correctness of the final answer. The ability to accurately derive and apply these trigonometric relationships is crucial for success in more advanced mathematical analyses and applications.

Now that we have sin(a), cos(a), sin(b), and cos(b), we can find sin(a + b) using the formula:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Substituting the values we found:

sin(a + b) = (-2/5)(1/3) + (√21 / 5)(-2√2 / 3)

sin(a + b) = -2/15 - (2√42) / 15

sin(a + b) = (-2 - 2√42) / 15

The final calculation brings together all the previously derived values into a single, cohesive step. The successful substitution and simplification showcase a firm grasp of algebraic manipulation combined with trigonometric principles. The resultant expression, sin(a + b) = (-2 - 2√42) / 15, represents the solution to the original problem. This answer encapsulates the accurate application of trigonometric identities and the understanding of quadrant rules. It underscores the importance of precision and attention to detail throughout the problem-solving process. This result is not only a numerical answer but also a testament to the logical and mathematical reasoning applied in solving the problem.

Therefore, sin(a + b) = (-2 - 2√42) / 15.

In summary, to find sin(a + b), we first used the Pythagorean identity to find cos(a) and sin(b), considering the quadrant in which the angles lie to determine the correct signs. Then, we applied the formula for sin(a + b) to obtain the final result. This problem illustrates the importance of understanding trigonometric identities and quadrant rules in solving trigonometric equations. Mastering these concepts allows for efficient and accurate problem-solving in various mathematical contexts. The solution to this problem not only provides a numerical answer but also reinforces the critical thinking and problem-solving skills necessary for advanced mathematical studies. The systematic approach, from identifying the necessary trigonometric identities to meticulously calculating each component, highlights the interconnectedness of mathematical concepts and their application in practical scenarios.

Trigonometry problem: How to find sin(a+b) given sin(a) and cos(b) in the fourth quadrant?