Question

find the exact value of each of the following under the given conditions.
a. \\( \cos ( \alpha + \beta ) \\) b. \\( \sin ( \alpha + \beta ) \\) c. \\( \tan ( \alpha + \beta ) \\)
\\( \tan \alpha = \frac { 3 } { 4 } , \pi < \alpha < \frac { 3 \pi } { 2 } \\), and \\( \cos \beta = \frac { 2 } { \sqrt { 13 } } , \frac { 3 \pi } { 2 } < \beta < 2 \pi \\)

a. \\( \cos ( \alpha + \beta ) = - \frac { 17 \sqrt { 13 } } { 65 } \\)
(type an exact answer using radicals as needed. simplify your answer. rationalize all denominators. use integers or
fractions for any numbers in the expression.)
b. \\( \sin ( \alpha + \beta ) = \frac { 6 \sqrt { 13 } } { 65 } \\)
(type an exact answer using radicals as needed. simplify your answer. rationalize all denominators. use integers or
fractions for any numbers in the expression.)
c. \\( \tan ( \alpha + \beta ) = \square \\)
(type an exact answer using radicals as needed. simplify your answer. rationalize all denominators. use integers or
fractions for any numbers in the expression.)

Explanation:

Step1: Recall the formula for \(\tan(A + B)\)

The formula for \(\tan(A + B)=\frac{\sin(A + B)}{\cos(A + B)}\)

Step2: Substitute the values of \(\sin(\alpha+\beta)\) and \(\cos(\alpha+\beta)\)

We know that \(\sin(\alpha+\beta)=\frac{6\sqrt{13}}{65}\) and \(\cos(\alpha+\beta)=-\frac{17\sqrt{13}}{65}\)

So, \(\tan(\alpha+\beta)=\frac{\frac{6\sqrt{13}}{65}}{-\frac{17\sqrt{13}}{65}}\)

Step3: Simplify the expression

\(\tan(\alpha+\beta)=\frac{6\sqrt{13}}{65}\times\frac{65}{- 17\sqrt{13}}=-\frac{6}{17}\)

Answer:

\(-\frac{6}{17}\)