use the given information to find (a) sin (s + t), (b) tan (s + t), and (c) the quadrant of s + t. cos s =…

Explanation:

Step1: Find (\sin s) and (\cos t)

Since (\sin^{2}\alpha+\cos^{2}\alpha = 1), for (s) with (\cos s=\frac{12}{13}) (in quadrant I), we have (\sin s=\sqrt{1 - \cos^{2}s}=\sqrt{1-\left(\frac{12}{13}\right)^{2}}=\sqrt{\frac{169 - 144}{169}}=\sqrt{\frac{25}{169}}=\frac{5}{13}). For (t) with (\sin t=\frac{3}{5}) (in quadrant I), we have (\cos t=\sqrt{1-\sin^{2}t}=\sqrt{1 - \left(\frac{3}{5}\right)^{2}}=\sqrt{\frac{25-9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}).

Step2: Use the sum - formula for sine (\sin(A + B)=\sin A\cos B+\cos A\sin B)

Here (A = s) and (B=t), so (\sin(s + t)=\sin s\cos t+\cos s\sin t). Substitute (\sin s=\frac{5}{13}), (\cos t=\frac{4}{5}), (\cos s=\frac{12}{13}), and (\sin t=\frac{3}{5}) into the formula: (\sin(s + t)=\frac{5}{13}\times\frac{4}{5}+\frac{12}{13}\times\frac{3}{5}=\frac{20 + 36}{65}=\frac{56}{65}).

Answer:

(\frac{56}{65})