Given that \(\sin x = \frac{5}{13}\) and \(\sin y = \frac{8}{17}\), where x and y are acute, find \(\cos(x+y)\).
A. \(\frac{130}{221}\) B. \(\frac{140}{221}\) C. \(\frac{140}{204}\) D. \(\frac{220}{23}\)
Correct Answer: B
Explanation
\(\cos(x+y) = \cos x\cos y - \sin x\sin y\)
Given \(\sin\) of an angle implies we have the value of the opposite and hypotenuse of the right-angled triangle. We find the adjacent side using Pythagoras' theorem.
\(Adj^{2} = Hyp^{2} - Opp^{2}\)
For triangle with angle x, \(adj = \sqrt{13^{2} - 5^{2}} = \sqrt{144} = 12\)
For triangle with angle y, \(adj = \sqrt{17^{2} - 8^{2}} = \sqrt{225} = 15\)
\(\therefore \cos x = \frac{12}{13}; \cos y = \frac{15}{17}\)