Finding Cos(A+B) Given Sin(A) And Sin(B) In Specific Quadrants

In trigonometry, finding the values of trigonometric functions for compound angles is a common problem. This article delves into a specific case where we are given the values of $\sin(A)$ and $\sin(B)$, along with the quadrants in which angles $A$ and $B$ lie. Our goal is to find the value of $\cos(A+B)$. This problem requires a solid understanding of trigonometric identities, quadrant rules, and algebraic manipulation. We will break down the problem step-by-step, providing a clear and comprehensive solution. The key to success in trigonometry lies in mastering the fundamental identities and understanding how the signs of trigonometric functions vary across different quadrants. Let's embark on this trigonometric journey to find $\cos(A+B)$. This exploration will not only solve the specific problem but also enhance your problem-solving skills in trigonometry. By the end of this discussion, you'll be equipped with the knowledge to tackle similar problems involving compound angles and trigonometric identities. Remember, practice is paramount in mathematics, so working through examples like this is crucial for mastering the concepts.

Problem Statement

We are given that $\sin(A) = \frac{12}{13}$, where $A$ is in the second quadrant, and $\sin(B) = -\frac{15}{17}$, where $B$ is in the third quadrant. Our task is to find the value of $\cos(A+B)$ and express it as a simplified fraction. This problem combines the concepts of trigonometric identities, quadrant rules, and algebraic manipulation. To solve this, we'll need to use the cosine addition formula, which states that $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$. We already have the values of $\sin(A)$ and $\sin(B)$, but we need to find $\cos(A)$ and $\cos(B)$. This is where the quadrant information becomes crucial, as it will help us determine the signs of the cosine values. Remember, in the second quadrant, cosine is negative, and in the third quadrant, both sine and cosine are negative. With this knowledge, we can use the Pythagorean identity to find the cosine values and then substitute everything into the addition formula. The final step is to simplify the resulting expression to obtain the answer as a simplified fraction. This problem is a great example of how different trigonometric concepts come together to solve a complex problem. Let's proceed step by step to unravel the solution.

Finding cos(A) and cos(B)

To find $\cos(A)$ and $\cos(B)$, we will use the Pythagorean identity, which states that $\sin^2(x) + \cos^2(x) = 1$. First, let's find $\cos(A)$. We know that $\sin(A) = \frac{12}{13}$, so $\sin^2(A) = \left(\frac{12}{13}\right)^2 = \frac{144}{169}$. Using the Pythagorean identity, we have $\cos^2(A) = 1 - \sin^2(A) = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169}$. Taking the square root of both sides, we get $\cos(A) = \pm\frac{5}{13}$. Since $A$ is in the second quadrant, where cosine is negative, we have $\cos(A) = -\frac{5}{13}$. Now, let's find $\cos(B)$. We know that $\sin(B) = -\frac{15}{17}$, so $\sin^2(B) = \left(-\frac{15}{17}\right)^2 = \frac{225}{289}$. Using the Pythagorean identity, we have $\cos^2(B) = 1 - \sin^2(B) = 1 - \frac{225}{289} = \frac{289 - 225}{289} = \frac{64}{289}$. Taking the square root of both sides, we get $\cos(B) = \pm\frac{8}{17}$. Since $B$ is in the third quadrant, where cosine is also negative, we have $\cos(B) = -\frac{8}{17}$. We have now successfully found both $\cos(A)$ and $\cos(B)$, which are crucial for applying the cosine addition formula. The careful application of the Pythagorean identity, combined with the quadrant rules, has allowed us to determine the correct signs and values for these trigonometric functions. With these values in hand, we are well-prepared to calculate $\cos(A+B)$.

Applying the Cosine Addition Formula

Now that we have found the values of $\cos(A)$ and $\cos(B)$, we can use the cosine addition formula to find $\cos(A+B)$. The formula is given by: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ We have: * $\sin(A) = \frac{12}{13}$ * $\cos(A) = -\frac{5}{13}$ * $\sin(B) = -\frac{15}{17}$ * $\cos(B) = -\frac{8}{17}$ Substituting these values into the formula, we get: $\cos(A+B) = \left(-\frac5}{13}\right)\left(-\frac{8}{17}\right) - \left(\frac{12}{13}\right)\left(-\frac{15}{17}\right)$ Now, we perform the multiplication $\cos(A+B) = \frac{40221} + \frac{180}{221}$ Adding the fractions, we get $\cos(A+B) = \frac{40 + 180{221} = \frac{220}{221}$ Thus, the value of $\cos(A+B)$ is $\frac{220}{221}$. This result is a simplified fraction, as 220 and 221 have no common factors other than 1. The successful application of the cosine addition formula, combined with the previously calculated values of $\cos(A)$ and $\cos(B)$, has led us to the final answer. This problem showcases the power of trigonometric identities in simplifying complex expressions and solving trigonometric problems. The ability to correctly apply these identities is a fundamental skill in trigonometry. With this solution, we have demonstrated a clear and concise approach to finding the cosine of a compound angle given the sine values of the individual angles and their respective quadrants.

Final Answer

Therefore, we have found that $\cos(A+B) = \frac{220}{221}$. This is the final answer, expressed as a simplified fraction. This solution demonstrates a comprehensive understanding of trigonometric identities, quadrant rules, and algebraic manipulation. We started with the given information about $\sin(A)$ and $\sin(B)$ and their respective quadrants. We then used the Pythagorean identity to find $\cos(A)$ and $\cos(B)$, carefully considering the signs based on the quadrants. Finally, we applied the cosine addition formula and simplified the result to obtain the final answer. The entire process highlights the interconnectedness of different trigonometric concepts and the importance of a systematic approach to problem-solving. Mastering these techniques is essential for success in trigonometry and related fields. This problem serves as a valuable exercise in applying trigonometric principles and reinforces the importance of accuracy and attention to detail in mathematical calculations. We have successfully navigated the complexities of this problem and arrived at a clear and concise solution.