Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 24 } { 25 } , \pi

Question

Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 24 } { 25 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \tan \beta = - \frac { 2 \sqrt { 21 } } { 21 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$
A) $\frac { 48 - 7 \sqrt { 21 } } { 125 }$

B) $\frac { 48 + 7 \sqrt { 21 } } { 125 }$

C) $\frac { 14 - 24 \sqrt { 21 } } { 125 }$

D) $\frac { - 14 - 24 \sqrt { 21 } } { 125 }$

Quizplus · Accepted Answer

Given $\sin \alpha = -\frac{24}{25}$, in the third quadrant both sine and cosine are negative, so $\cos \alpha = -\frac{7}{25}$. For $\tan \beta = -\frac{2\sqrt{21}}{21}$, in the second quadrant sine is positive and cosine is negative, so $\sin \beta = \frac{2\sqrt{21}}{\sqrt{441}} = \frac{2\sqrt{21}}{21}$ and $\cos \beta = -\frac{\sqrt{441 - 84}}{21} = -\frac{\sqrt{357}}{21} = -\frac{3\sqrt{21}}{21}$. Using the cosine addition formula, $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta = \left(-\frac{7}{25}\right)\left(-\frac{3\sqrt{21}}{21}\right) - \left(-\frac{24}{25}\right)\left(\frac{2\sqrt{21}}{21}\right) = \frac{21\sqrt{21} + 48\sqrt{21}}{525} = \frac{48 + 7\sqrt{21}}{125}$.

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