Find the exact value under the given conditions.
-$\sin \alpha = \frac { 7 } { 25 } , 0

Question

Find the exact value under the given conditions.
-$\sin \alpha = \frac { 7 } { 25 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \cos \beta = \frac { 20 } { 29 } , 0 < \beta < \frac { \pi } { 2 } \quad$ Find $\cos ( \alpha + \beta )$.
A) $\frac { 333 } { 725 }$

B) $\frac { 644 } { 725 }$

C) $- \frac { 364 } { 725 }$
 
D) $\frac { 627 } { 725 }$

Quizplus · Accepted Answer

The formula for $\cos(\alpha + \beta)$ is $\cos\alpha\cos\beta - \sin\alpha\sin\beta$. Given $\sin\alpha = \frac{7}{25}$, we find $\cos\alpha = \sqrt{1 - \sin^2\alpha} = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \frac{24}{25}$. Given $\cos\beta = \frac{20}{29}$, we find $\sin\beta = \sqrt{1 - \cos^2\beta} = \sqrt{1 - \left(\frac{20}{29}\right)^2} = \frac{21}{29}$. Plugging these into the formula gives $\frac{24}{25} \cdot \frac{20}{29} - \frac{7}{25} \cdot \frac{21}{29} = \frac{480}{725} - \frac{147}{725} = \frac{333}{725}$.

Find the Exact Value Under the Given Conditions sin⁡α=725,0<α<π2;cos⁡β=2029,0<β<π2\sin \alpha = \frac { 7 } { 25 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \cos \beta = \frac { 20 } { 29 } , 0 < \beta < \frac { \pi } { 2 } \quadsinα=257​,0<α<2π​;cosβ=2920​,0<β<2π​

Find the Exact Value Under the Given Conditions $\sin \alpha = \frac { 7 } { 25 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \cos \beta = \frac { 20 } { 29 } , 0 < \beta < \frac { \pi } { 2 } \quad$