Question

use the given information to find (a) \\( \sin (s+t) \\), (b) \\( \tan (s+t) \\), and (c) the quadrant of \\( s+t \\).
\\( \cos s=-\frac{3}{5} \\) and \\( \cos t=-\frac{8}{17} \\), \\( s \\) and \\( t \\) in quadrant ii
(a) \\( \sin (s+t)= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
(b) \\( \tan (s+t)= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
(c) what is the quadrant of \\( s+t \\)?
quadrant iv
quadrant iii
quadrant
quadrant

Explanation:

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

Since \(\sin^{2}\alpha+\cos^{2}\alpha = 1\), for \(s\) with \(\cos s=-\frac{3}{5}\) (in quadrant II, \(\sin s>0\)):
\(\sin s=\sqrt{1-\cos^{2}s}=\sqrt{1 - (-\frac{3}{5})^{2}}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}\)
For \(t\) with \(\cos t =-\frac{8}{17}\) (in quadrant II, \(\sin t>0\)):
\(\sin t=\sqrt{1-\cos^{2}t}=\sqrt{1-(-\frac{8}{17})^{2}}=\sqrt{1-\frac{64}{289}}=\sqrt{\frac{225}{289}}=\frac{15}{17}\)

Step2: Calculate \(\sin(s + t)\)

Using the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\)
\(\sin(s + t)=\sin s\cos t+\cos s\sin t\)
Substitute \(\sin s=\frac{4}{5},\cos s=-\frac{3}{5},\sin t=\frac{15}{17},\cos t=-\frac{8}{17}\)
\(\sin(s + t)=\frac{4}{5}\times(-\frac{8}{17})+(-\frac{3}{5})\times\frac{15}{17}\)
\(=\frac{-32 - 45}{85}=\frac{-77}{85}\)

Step3: Calculate \(\tan s\) and \(\tan t\)

\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\), so \(\tan s=\frac{\sin s}{\cos s}=\frac{\frac{4}{5}}{-\frac{3}{5}}=-\frac{4}{3}\)
\(\tan t=\frac{\sin t}{\cos t}=\frac{\frac{15}{17}}{-\frac{8}{17}}=-\frac{15}{8}\)

Step4: Calculate \(\tan(s + t)\)

Using the formula \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\)
\(\tan(s + t)=\frac{-\frac{4}{3}+(-\frac{15}{8})}{1-(-\frac{4}{3})\times(-\frac{15}{8})}\)
First, simplify the numerator: \(-\frac{4}{3}-\frac{15}{8}=-\frac{32 + 45}{24}=-\frac{77}{24}\)
Simplify the denominator: \(1-\frac{60}{24}=1-\frac{5}{2}=-\frac{3}{2}\)
Then \(\tan(s + t)=\frac{-\frac{77}{24}}{-\frac{3}{2}}=\frac{77}{36}\)

Step5: Determine the quadrant of \(s + t\)

Since \(\sin(s + t)<0\) and \(\tan(s + t)>0\)
In quadrant III, \(\sin\theta<0\) and \(\tan\theta>0\)

Answer:

(a) \(\sin(s + t)=-\frac{77}{85}\)
(b) \(\tan(s + t)=\frac{77}{36}\)
(c) Quadrant III