Use the given information to find the exact value of the expression.
-$\sin \alpha = - \frac { 24 } { 25 } , \alpha$ lies in quadrant IV, and $\cos \beta = - \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant III Find $\sin ( \alpha - \beta )$.
A) $\frac { 14 + 24 \sqrt { 21 } } { 125 }$
B) $\frac { - 14 + 24 \sqrt { 21 } } { 125 }$
C) $\frac { - 48 + 7 \sqrt { 21 } } { 125 }$
D) $\frac { - 48 - 7 \sqrt { 21 } } { 125 }$

Question

Quizplus · Accepted Answer

The formula for $\sin(\alpha - \beta)$ is $\sin \alpha \cos \beta - \cos \alpha \sin \beta$. Given $\sin \alpha = -\frac{24}{25}$ and $\cos \beta = -\frac{\sqrt{21}}{5}$, we need to find $\cos \alpha$ and $\sin \beta$. Since $\alpha$ is in quadrant IV, $\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(-\frac{24}{25}\right)^2} = \frac{7}{25}$. Because $\beta$ is in quadrant III, $\sin \beta = \sqrt{1 - \cos^2 \beta} = \sqrt{1 - \left(-\frac{\sqrt{21}}{5}\right)^2} = -\frac{2}{5}$. Substituting these values into the formula gives $\sin(\alpha - \beta) = \left(-\frac{24}{25}\right)\left(-\frac{\sqrt{21}}{5}\right) - \left(\frac{7}{25}\right)\left(-\frac{2}{5}\right) = \frac{24\sqrt{21}}{125} + \frac{14}{125} = \frac{14 + 24\sqrt{21}}{125}$.

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