Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 3 } { 5 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant I $\quad$ Find $\cos ( \alpha - \beta )$.
A) $\frac { - 8 + 3 \sqrt { 21 } } { 25 }$
B) $\frac { 6 + 4 \sqrt { 21 } } { 25 }$
C) $\frac { 6 - 4 \sqrt { 21 } } { 25 }$
D) $\frac { 8 - 3 \sqrt { 21 } } { 25 }$

Question

Quizplus · Accepted Answer

The formula for $\cos(\alpha - \beta)$ is $\cos \alpha \cos \beta + \sin \alpha \sin \beta$. Given $\sin \alpha = \frac{3}{5}$ and $\cos \beta = \frac{2}{5}$, we find $\cos \alpha$ and $\sin \beta$ using Pythagoras' theorem. Since $\alpha$ is in quadrant II, $\cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\frac{4}{5}$. For $\beta$ in quadrant I, $\sin \beta = \sqrt{1 - \cos^2 \beta} = \sqrt{1 - \left(\frac{2}{5}\right)^2} = \frac{\sqrt{21}}{5}$. Thus, $\cos(\alpha - \beta) = \left(-\frac{4}{5}\right)\left(\frac{2}{5}\right) + \left(\frac{3}{5}\right)\left(\frac{\sqrt{21}}{5}\right) = \frac{-8 + 3\sqrt{21}}{25}$.

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