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

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:

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Answer:

$-\frac{84}{85}$