Use the given information to find $(a) \sin (s+t),(b) \tan (s+t)$, and $(c)$ the quadrant of $s+t$.
\[
\cos s=\frac{8}{17} \text { and } \cos t=\frac{4}{5} \text {, s and } t \text { in quadrant } I
\]
(a) $\sin (s+t)=$
(Use integers or fractions for any numbers in the expression.)

Step 1 ：We are given that \(\cos s = \frac{8}{17}\) and \(\cos t = \frac{4}{5}\), and that both \(s\) and \(t\) are in quadrant I.

Step 2 ：We need to find \(\sin s\) and \(\sin t\) to calculate \(\sin (s+t)\). Since \(s\) and \(t\) are in quadrant I, both \(\sin s\) and \(\sin t\) are positive.

Step 3 ：We can use the Pythagorean identity \(\sin^2 x = 1 - \cos^2 x\) to find \(\sin s\) and \(\sin t\).

Step 4 ：Using the Pythagorean identity, we find that \(\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(\frac{8}{17}\right)^2} = \frac{15}{17}\) and \(\sin t = \sqrt{1 - \cos^2 t} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}\).

Step 5 ：We know that \(\sin (s+t) = \sin s \cos t + \cos s \sin t\). Substituting the values we found, we get \(\sin (s+t) = \left(\frac{15}{17}\right) \left(\frac{4}{5}\right) + \left(\frac{8}{17}\right) \left(\frac{3}{5}\right) = \frac{60}{85} + \frac{24}{85} = \frac{84}{85}\).

Step 6 ：So, \(\sin (s+t) = \boxed{\frac{84}{85}}\)