Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 7 } { 25 }$ and $\cos v = - \frac { 12 } { 13 }$ .(Both u and v are in Quadrant II. )
A) $\sin ( u + v ) = \frac { 36 } { 325 }$
B) $\sin ( u + v ) = - \frac { 36 } { 325 }$
C) $\sin ( u + v ) = - \frac { 204 } { 325 }$
D) $\sin ( u + v ) = - \frac { 253 } { 325 }$
E) $\sin ( u + v ) = \frac { 204 } { 325 }$

Question

Quizplus · Accepted Answer

To find $\sin(u + v)$, we use the identity $\sin(u + v) = \sin u \cos v + \cos u \sin v$. Given $\sin u = \frac{7}{25}$, we can find $\cos u$ using the Pythagorean identity $\sin^2 u + \cos^2 u = 1$, which gives $\cos u = \pm\sqrt{1 - \left(\frac{7}{25}\right)^2} = \pm\frac{24}{25}$. Since $u$ is in Quadrant II, $\cos u = -\frac{24}{25}$. Given $\cos v = -\frac{12}{13}$, we find $\sin v$ similarly: $\sin v = \pm\sqrt{1 - \left(-\frac{12}{13}\right)^2} = \pm\frac{5}{13}$. In Quadrant II, $\sin v$ is positive, so $\sin v = \frac{5}{13}$. Thus, $\sin(u + v) = \left(\frac{7}{25}\right)\left(-\frac{12}{13}\right) + \left(-\frac{24}{25}\right)\left(\frac{5}{13}\right) = -\frac{84}{325} - \frac{120}{325} = -\frac{204}{325}$.

Find the Exact Value Of sin⁡(u+v)\sin ( u + v )sin(u+v)

Find the Exact Value Of $\sin ( u + v )$