Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 24 } { 25 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos ( \alpha - \beta )$.
A) $\frac { - 14 + 24 \sqrt { 21 } } { 125 }$
B) $\frac { 48 + 7 \sqrt { 21 } } { 125 }$
C) $\frac { 48 - 7 \sqrt { 21 } } { 125 }$
D) $\frac { 14 - 24 \sqrt { 21 } } { 125 }$

Question

Quizplus · Accepted Answer

The formula for $\cos(\alpha - \beta)$ is $\cos\alpha\cos\beta + \sin\alpha\sin\beta$. Since $\sin\alpha = \frac{24}{25}$ and $\alpha$ is in quadrant II, $\cos\alpha = -\sqrt{1-\sin^2\alpha} = -\sqrt{1-\left(\frac{24}{25}\right)^2} = -\frac{7}{25}$. For $\beta$ in quadrant I, where $\cos\beta = \frac{2}{5}$, $\sin\beta = \sqrt{1-\cos^2\beta} = \sqrt{1-\left(\frac{2}{5}\right)^2} = \frac{\sqrt{21}}{5}$. Substituting these values in, we get $\cos(\alpha - \beta) = -\frac{7}{25} \cdot \frac{2}{5} + \frac{24}{25} \cdot \frac{\sqrt{21}}{5} = \frac{-14 + 24\sqrt{21}}{125}$.

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