Given that \(sin x = \frac{4}{5}\) and \(cos y = \frac{12}{13}\), where x is an obtuse angle and y is an acute angle, find the value of sin (x – y).
-
\(\frac{63}{65}\)
- \(\frac{48}{65}\)
- \(\frac{56}{65}\)
- \(\frac{16}{65}\)
The correct answer is: A
Explanation
\(sin x = \frac{4}{5}\) and \(cos y = \frac{12}{13}\)
x is obtuse i.e sin x = + ve while cos x = + ve
\(cos x=\frac{3}{5}==>cos x=-\frac {3}{5}(obtuse)\)
\(sin y= \frac{5}{13}\)
\(sin (x-y) = sin x\) \(cos y - cos x\) \(sin y\)
\(sin(x-y) = \frac{4}{5}\times\frac{12}{13}-(-\frac{3}{5})\times\frac{5}{13}\)
\(sin(x-y) = \frac{48}{65}-(-\frac{3}{13})\)
\(\therefore sin (x-y) = \frac{48}{65} + \frac{3}{13} = \frac{63}{65}\)