Find the exact value of the expression using the provided information.
-Find $\sin ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 15 } { 17 }$, with $t$ in quadrant II.
A) $\frac { 21 } { 221 }$
B) $- \frac { 21 } { 221 }$
C) $- \frac { 171 } { 221 }$
D) $\frac { 171 } { 221 }$

Question

Quizplus · Accepted Answer

To find $\sin(s + t)$, we use the identity $\sin(s + t) = \sin(s)\cos(t) + \cos(s)\sin(t)$. Given $\cos(s) = -\frac{5}{13}$ and $\sin(t) = \frac{15}{17}$, we need to find $\sin(s)$ and $\cos(t)$. Since $s$ is in quadrant II, $\sin(s) = \sqrt{1 - \cos^2(s)} = \sqrt{1 - \left(-\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}$. For $t$ in quadrant II, $\cos(t) = -\sqrt{1 - \sin^2(t)} = -\sqrt{1 - \left(\frac{15}{17}\right)^2} = -\sqrt{1 - \frac{225}{289}} = -\sqrt{\frac{64}{289}} = -\frac{8}{17}$. Thus, $\sin(s + t) = \left(\frac{12}{13}\right)\left(-\frac{8}{17}\right) + \left(-\frac{5}{13}\right)\left(\frac{15}{17}\right) = -\frac{96}{221} - \frac{75}{221} = -\frac{171}{221}$.

Find the Exact Value of the Expression Using the Provided sin⁡(s+t)\sin ( s + t )sin(s+t)

Find the Exact Value of the Expression Using the Provided $\sin ( s + t )$