Question

find the exact value of the expressions $\cos(\alpha+\beta)$, $\sin(\alpha+\beta)$ and $\tan(\alpha+\beta)$ under the
$\sin(\alpha)=\frac{24}{25}$, $\alpha$ lies in quadrant i, and $\sin(\beta)=\frac{15}{17}$, $\beta$ lies in quadrant ii.

a. $\cos(\alpha+\beta)=\square$
(simplify your answer. type an exact answer, using radicals as needed. use integers o
the expression.)

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 \(\cos\alpha>0\)):
\(\cos\alpha=\sqrt{1-\sin^{2}\alpha}=\sqrt{1 - (\frac{24}{25})^{2}}=\sqrt{\frac{625 - 576}{625}}=\sqrt{\frac{49}{625}}=\frac{7}{25}\)

Step2: Find \(\cos\beta\)

Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), for \(\beta\) with \(\sin\beta=\frac{15}{17}\) (in quadrant II where \(\cos\beta<0\)):
\(\cos\beta=-\sqrt{1-\sin^{2}\beta}=-\sqrt{1 - (\frac{15}{17})^{2}}=-\sqrt{\frac{289 - 225}{289}}=-\sqrt{\frac{64}{289}}=-\frac{8}{17}\)

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

Here \(A=\alpha\) and \(B = \beta\), so \(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\)
Substitute \(\cos\alpha=\frac{7}{25}\), \(\cos\beta=-\frac{8}{17}\), \(\sin\alpha=\frac{24}{25}\), \(\sin\beta=\frac{15}{17}\)
\(\cos(\alpha+\beta)=\frac{7}{25}\times(-\frac{8}{17})-\frac{24}{25}\times\frac{15}{17}\)
\(=\frac{-56 - 360}{425}=\frac{-416}{425}\)

Answer:

\(\frac{- 416}{425}\)