Compute $\sin ( \alpha - \beta )$ and $\cos ( \alpha - \beta )$ using the data below. $\sin \alpha = \frac { 12 } { 13 } \quad \text { where } \frac { \pi } { 2 }

Question

Compute $\sin ( \alpha - \beta )$ and $\cos ( \alpha - \beta )$ using the data below. $\sin \alpha = \frac { 12 } { 13 } \quad \text { where } \frac { \pi } { 2 } < \alpha < \pi$ $$\cos \beta = - \frac { 15 } { 17 } \quad \text { where } \pi < \beta < \frac { 3 \pi } { 2 }$$
A) $\sin ( \alpha - \beta ) = \frac { 140 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 171 } { 221 }$
B) $\sin ( \alpha - \beta ) = - \frac { 220 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 21 } { 221 }$
C) $\sin ( \alpha - \beta ) = - \frac { 171 } { 221 }$ $\cos ( \alpha - \beta ) = \frac { 140 } { 221 }$
D) $\sin ( \alpha - \beta ) = \frac { 220 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 21 } { 221 }$
E) $\sin ( \alpha - \beta ) = - \frac { 21 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 220 } { 221 }$

Quizplus · Accepted Answer

Using the identities $\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta$ and $\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta$, and calculating $\cos\alpha = -\frac{5}{13}$ and $\sin\beta = -\frac{8}{17}$, we find $\sin(\alpha - \beta) = -\frac{220}{221}$ and $\cos(\alpha - \beta) = -\frac{21}{221}$.

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