Question

find the exact value of the expressions \\( \cos ( \alpha + \beta ) , \sin ( \alpha + \beta ) \\) and \\( \tan ( \alpha + \beta ) \\) under the following conditions:
\\( \sin ( \alpha ) = \frac { 12 } { 13 } , \alpha \\) lies in quadrant i, and \\( \sin ( \beta ) = \frac { 4 } { 5 } , \beta \\) lies in quadrant ii.

\\( \cos ( \alpha + \beta ) = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression )

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 \(\cos\alpha>0\)):

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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 \(\cos\beta<0\)):

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

Here \(A=\alpha\) and \(B = \beta\), so:

$$
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Answer:

\(-\frac{63}{65}\)