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

use the given information to find (a) \\( \\sin (s+t) \\), (b) \\( \\tan (s+t) \\), and (c) the quadrant of \\( s+t \\).\n\\( \\cos s=\\frac{8}{17} \\) and \\( \\cos t=\\frac{4}{5} \\), \\( s \\) and \\( t \\) in quadrant iv\n(a) \\( \\sin (s+t)= \\) \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

use the given information to find (a) \\( \\sin (s+t) \\), (b) \\( \\tan (s+t) \\), and (c) the quadrant of \\( s+t \\).\n\\( \\cos s=\\frac{8}{17} \\) and \\( \\cos t=\\frac{4}{5} \\), \\( s \\) and \\( t \\) in quadrant iv\n(a) \\( \\sin (s+t)= \\) \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find $\sin s$ and $\sin t$

Since $\sin^{2}\alpha+\cos^{2}\alpha = 1$, for $\alpha=s$: $\sin^{2}s=1-\cos^{2}s=1 - (\frac{8}{17})^{2}=1-\frac{64}{289}=\frac{289 - 64}{289}=\frac{225}{289}$, and since $s$ is in quadrant IV, $\sin s=-\frac{15}{17}$. For $\alpha = t$: $\sin^{2}t=1-\cos^{2}t=1-(\frac{4}{5})^{2}=1-\frac{16}{25}=\frac{9}{25}$, and since $t$ is in quadrant IV, $\sin t=-\frac{3}{5}$.

Step2: Use the sum - formula for sine

The sum - formula for sine is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A = s$ and $B = t$, so $\sin(s + t)=\sin s\cos t+\cos s\sin t$. Substitute $\sin s=-\frac{15}{17},\cos s=\frac{8}{17},\sin t=-\frac{3}{5},\cos t=\frac{4}{5}$ into the formula: [ \begin{align*} \sin(s + t)&=(-\frac{15}{17})\times\frac{4}{5}+\frac{8}{17}\times(-\frac{3}{5})\ &=-\frac{60}{85}-\frac{24}{85}\ &=-\frac{60 + 24}{85}\ &=-\frac{84}{85} \end{align*} ]

Answer:

$-\frac{84}{85}$