Given that \(\sin x = \frac{5}{13}\) and \(\sin y = \frac{8}{17}\), where x and y are acute, find \(\cos(x+y)\).

Answer Details

To solve the problem, we need to use the sum formula for cosine: \[\cos(x+y) = \cos x \cos y - \sin x \sin y\] We are given the values of \(\sin x\) and \(\sin y\), so we need to find \(\cos x\) and \(\cos y\). To do this, we use the fact that \(\sin^2 x + \cos^2 x = 1\) and \(\sin^2 y + \cos^2 y = 1\) for any angle x and y. From \(\sin x = \frac{5}{13}\), we can find \(\cos x\) by using the Pythagorean identity: \[\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}\] Similarly, from \(\sin y = \frac{8}{17}\), we can find \(\cos y\): \[\cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - \left(\frac{8}{17}\right)^2} = \frac{15}{17}\] Now we can substitute into the formula for \(\cos(x+y)\): \[\cos(x+y) = \cos x \cos y - \sin x \sin y = \frac{12}{13} \cdot \frac{15}{17} - \frac{5}{13} \cdot \frac{8}{17} = \frac{140}{221}\] Therefore, the answer is \(\frac{140}{221}\).