find the exact value of the expressions $ \cos ( \alpha + \beta ) $, $ \sin ( \alpha + \beta ) $ and $ \tan ( \alpha + \beta ) $ under the following conditions: $ \sin ( \alpha ) = \frac { 24 } { 25 } $, $ \alpha $ lies in quadrant i, and $ \sin ( \beta ) = \frac { 4 } { 5 } $, $ \beta $ lies in quadrant ii. a. $ \cos ( \alpha + \beta ) = $ (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Question

Accepted Answer

# Explanation:
## Step1: Find $\cos\alpha$
Using the identity $\sin^{2}\theta+\cos^{2}\theta = 1$, for $\alpha$ with $\sin\alpha=\frac{24}{25}$ (in quadrant I where $\cos\alpha>0$):
$$
\begin{align*}
\cos\alpha&=\sqrt{1-\sin^{2}\alpha}\
&=\sqrt{1 - (\frac{24}{25})^{2}}\
&=\sqrt{\frac{625 - 576}{625}}\
&=\sqrt{\frac{49}{625}}\
&=\frac{7}{25}
\end{align*}
$$
## Step2: Find $\cos\beta$
Using the identity $\sin^{2}\theta+\cos^{2}\theta = 1$, for $\beta$ with $\sin\beta=\frac{4}{5}$ (in quadrant II where $\cos\beta<0$):
$$
\begin{align*}
\cos\beta&=-\sqrt{1-\sin^{2}\beta}\
&=-\sqrt{1-(\frac{4}{5})^{2}}\
&=-\sqrt{\frac{25 - 16}{25}}\
&=-\sqrt{\frac{9}{25}}\
&=-\frac{3}{5}
\end{align*}
$$
## Step3: Use the cosine addition formula $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$
Substitute $\cos\alpha=\frac{7}{25}$, $\cos\beta =-\frac{3}{5}$, $\sin\alpha=\frac{24}{25}$, $\sin\beta=\frac{4}{5}$ into the formula:
$$
\begin{align*}
\cos(\alpha+\beta)&=\frac{7}{25}\times(-\frac{3}{5})-\frac{24}{25}\times\frac{4}{5}\
&=\frac{-21 - 96}{125}\
&=\frac{-117}{125}
\end{align*}
$$

# Answer:
$\frac{-117}{125}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Find \(\cos\alpha\)

Step2: Find \(\cos\beta\)

Step3: Use the cosine addition formula \(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\)

Answer: