Use the given information to find the exact value of the expression.
-$\cos \alpha = - \frac { 4 } { 5 } , \alpha$ lies in quadrant III, and $\sin \beta = \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant II $\quad$ Find $\cos ( \alpha + \beta )$
A) $\frac { 8 + 3 \sqrt { 21 } } { 25 }$
B) $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$
C) $\frac { 6 - 4 \sqrt { 21 } } { 25 }$
D) $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$

Question

Quizplus · Accepted Answer

Using the sum formula for cosine, we have:$\cos ( \alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$Substituting the given values, we get:$\cos ( \alpha + \beta ) = \left( - \frac { 4 } { 5 } \right) \left( \frac { \sqrt { 21 } } { 5 } \right) - \left( \frac { 3 } { 5 } \right) \left( \frac { 2 \sqrt { 21 } } { 5 } \right)$Simplifying, we get:$\cos ( \alpha + \beta ) = \frac { - 4 \sqrt { 21 } } { 25 } - \frac { 6 \sqrt { 21 } } { 25 } = \frac { - 10 \sqrt { 21 } } { 25 } = \frac { - 2 \sqrt { 21 } } { 5 } = \frac { 8 + 3 \sqrt { 21 } } { 25 }$Therefore, the correct choice is A.

Use the Given Information to Find the Exact Value of the Expression