Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 5 } { 13 } , \frac { 3 \pi } { 2 }

Question

Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 5 } { 13 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \tan \beta = - \frac { 7 } { 24 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$.
A) $\frac { 204 } { 325 }$

B) $- \frac { 253 } { 325 }$

C) $\frac { 36 } { 325 }$

D) $- \frac { 323 } { 325 }$

Quizplus · Accepted Answer

In the fourth quadrant, $\cos \alpha = \frac{12}{13}$. In the second quadrant, $\cos \beta = -\frac{24}{25}$. Using the cosine addition formula, $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta = \frac{12}{13} \cdot -\frac{24}{25} - \left(-\frac{5}{13}\right) \cdot \frac{7}{25} = -\frac{288}{325} + \frac{35}{325} = -\frac{253}{325}$.

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