Find the exact value of $\sin (u-v)$ given that $\sin u=\frac{4}{5}$ and $\sin v=\frac{12}{13}$, with $u$ and $v$ in quadrant $I$.
Solution
Step 1 ：We are given that \(\sin u = \frac{4}{5}\) and \(\sin v = \frac{12}{13}\), with \(u\) and \(v\) in quadrant I.
Step 2 ：We need to find the exact value of \(\sin (u-v)\). The formula for \(\sin (u-v)\) is \(\sin u \cos v - \cos u \sin v\).
Step 3 ：We are not given the values of \(\cos u\) and \(\cos v\), but we can find these using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\).
Step 4 ：Calculating \(\cos u = \sqrt{1 - \sin^2 u}\) gives us \(\cos u = 0.5999999999999999\).
Step 5 ：Calculating \(\cos v = \sqrt{1 - \sin^2 v}\) gives us \(\cos v = 0.38461538461538447\).
Step 6 ：Substituting these values into the formula for \(\sin (u-v)\) gives us \(\sin (u-v) = \sin u \cos v - \cos u \sin v = -0.24615384615384617\).
Step 7 ：Thus, the exact value of \(\sin (u-v)\) is \(\boxed{-0.24615384615384617}\).
