use the cosine of a sum and cosine of a difference identities to find $ \cos (s+t) $ and $ \cos (s-t) $.
$ \sin s=-\frac{12}{13} $ and $ \sin t=\frac{3}{5} $, s in quadrant iii and t in quadrant i
$ \cos (s+t)= $
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
$ \cos (s-t)= $
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

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Accepted Answer

# Explanation:
## Step1: Find $\cos s$
Using the identity $\sin^{2}\alpha+\cos^{2}\alpha = 1$, for $s$ with $\sin s=-\frac{12}{13}$ (in quadrant III where $\cos s<0$):
$$
\begin{align*}
\cos^{2}s&=1-\sin^{2}s\
\cos^{2}s&=1 - (-\frac{12}{13})^{2}\
\cos^{2}s&=1-\frac{144}{169}\
\cos^{2}s&=\frac{169 - 144}{169}=\frac{25}{169}\
\cos s&=-\frac{5}{13}
\end{align*}
$$
## Step2: Find $\cos t$
Using the identity $\sin^{2}\alpha+\cos^{2}\alpha = 1$, for $t$ with $\sin t=\frac{3}{5}$ (in quadrant I where $\cos t>0$):
$$
\begin{align*}
\cos^{2}t&=1-\sin^{2}t\
\cos^{2}t&=1-(\frac{3}{5})^{2}\
\cos^{2}t&=1-\frac{9}{25}\
\cos^{2}t&=\frac{25 - 9}{25}=\frac{16}{25}\
\cos t&=\frac{4}{5}
\end{align*}
$$
## Step3: Use the cosine - of - a - sum formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$
Here $A = s$ and $B=t$, so $\cos(s + t)=\cos s\cos t-\sin s\sin t$
Substitute $\cos s=-\frac{5}{13}$, $\cos t=\frac{4}{5}$, $\sin s=-\frac{12}{13}$, $\sin t=\frac{3}{5}$
$$
\begin{align*}
\cos(s + t)&=(-\frac{5}{13})\times\frac{4}{5}-(-\frac{12}{13})\times\frac{3}{5}\
&=-\frac{20}{65}+\frac{36}{65}\
&=\frac{-20 + 36}{65}\
&=\frac{16}{65}
\end{align*}
$$
## Step4: Use the cosine - of - a - difference formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$
Here $A = s$ and $B = t$, so $\cos(s - t)=\cos s\cos t+\sin s\sin t$
Substitute $\cos s=-\frac{5}{13}$, $\cos t=\frac{4}{5}$, $\sin s=-\frac{12}{13}$, $\sin t=\frac{3}{5}$
$$
\begin{align*}
\cos(s - t)&=(-\frac{5}{13})\times\frac{4}{5}+(-\frac{12}{13})\times\frac{3}{5}\
&=-\frac{20}{65}-\frac{36}{65}\
&=\frac{-20-36}{65}\
&=-\frac{56}{65}
\end{align*}
$$

# Answer:
$\cos(s + t)=\frac{16}{65}$
$\cos(s - t)=-\frac{56}{65}$

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Question

Explanation:

Step1: Find \(\cos s\)

Step2: Find \(\cos t\)

Step3: Use the cosine - of - a - sum formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\)

Step4: Use the cosine - of - a - difference formula \(\cos(A - B)=\cos A\cos B+\sin A\sin B\)

Answer: