Given that (sin x = frac{5}{13}) and (sin y = frac{8}{17}), where x and y…

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

Correct Answer: Option B

Explanation

(cos(x+y) = cos xcos y – sin xsin 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})

(cos(x+y) = (frac{12}{13}timesfrac{15}{17}) – (frac{5}{13}timesfrac{8}{17}) = frac{180}{221} – frac{40}{221})

= (frac{140}{221})