Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 8 } { 17 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 12 } { 13 } , \beta$ lies in quadrant I $\quad$ Find $\sin ( \alpha - \beta )$.
A) $\frac { 171 } { 221 }$
B) $\frac { 220 } { 221 }$
C) $\frac { 140 } { 221 }$
D) $\frac { 21 } { 221 }$

Question

Quizplus · Accepted Answer

The sine difference formula is $\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$. Given $\sin\alpha = \frac{8}{17}$ and $\cos\beta = \frac{12}{13}$, we find $\cos\alpha$ and $\sin\beta$ using Pythagoras. For $\alpha$ in QII, $\cos\alpha = -\sqrt{1 - \sin^2\alpha} = -\frac{15}{17}$. For $\beta$ in QI, $\sin\beta = \sqrt{1 - \cos^2\beta} = \frac{5}{13}$. Substituting these values, $\sin(\alpha - \beta) = \frac{8}{17} \cdot \frac{12}{13} - (-\frac{15}{17}) \cdot \frac{5}{13} = \frac{96 + 75}{221} = \frac{171}{221}$.

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