find the exact value of the expressions \\( \\cos ( \\alpha + \\beta ) , \\sin ( \\alpha + \\beta ) \\) and…

Explanation:

Step1: Find (\cos\alpha)

Using the identity (\sin^{2}\theta+\cos^{2}\theta = 1), for (\alpha) with (\sin\alpha=\frac{12}{13}) (in quadrant I where (\cos\alpha>0)): [ \begin{align*} \cos\alpha&=\sqrt{1-\sin^{2}\alpha}\ &=\sqrt{1 - (\frac{12}{13})^{2}}\ &=\sqrt{1-\frac{144}{169}}\ &=\sqrt{\frac{169 - 144}{169}}\ &=\sqrt{\frac{25}{169}}\ &=\frac{5}{13} \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{1-\frac{16}{25}}\ &=-\sqrt{\frac{25 - 16}{25}}\ &=-\sqrt{\frac{9}{25}}\ &=-\frac{3}{5} \end{align*} ]

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

Here (A=\alpha) and (B = \beta), so: [ \begin{align*} \cos(\alpha+\beta)&=\cos\alpha\cos\beta-\sin\alpha\sin\beta\ &=\frac{5}{13}\times(-\frac{3}{5})-\frac{12}{13}\times\frac{4}{5}\ &=-\frac{15}{65}-\frac{48}{65}\ &=-\frac{63}{65} \end{align*} ]

Answer:

(-\frac{63}{65})