Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { \sqrt { 3 } } { 3 }$, with $\mathrm { s }$ in quadrant IV, and $\sin \mathrm { t } = - \frac { \sqrt { 5 } } { 6 }$, with $\mathrm { t }$ in quadrant
IV.
A) $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$

B) $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$

C) $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$

D) $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$

Question

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { \sqrt { 3 } } { 3 }$, with $\mathrm { s }$ in quadrant IV, and $\sin \mathrm { t } = - \frac { \sqrt { 5 } } { 6 }$, with $\mathrm { t }$ in quadrant
IV.
A) $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$

B) $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$

C) $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$

D) $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$

Quizplus · Accepted Answer

To find $\cos(s + t)$, we use the identity $\cos(s + t) = \cos s \cos t - \sin s \sin t$. Since $s$ and $t$ are in the fourth quadrant, both $\cos s$ and $\cos t$ are positive. Given $\sin s = -\frac{\sqrt{3}}{3}$ and $\sin t = -\frac{\sqrt{5}}{6}$, we find $\cos s$ and $\cos t$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$.For $s$, $\sin^2 s = \left(-\frac{\sqrt{3}}{3}\right)^2 = \frac{1}{3}$, so $\cos^2 s = 1 - \frac{1}{3} = \frac{2}{3}$. Thus, $\cos s = \sqrt{\frac{2}{3}}$.For $t$, $\sin^2 t = \left(-\frac{\sqrt{5}}{6}\right)^2 = \frac{5}{36}$, so $\cos^2 t = 1 - \frac{5}{36} = \frac{31}{36}$. Thus, $\cos t = \sqrt{\frac{31}{36}} = \frac{\sqrt{31}}{6}$.Substituting these into the identity gives:$$\cos(s + t) = \left(\sqrt{\frac{2}{3}}\right)\left(\frac{\sqrt{31}}{6}\right) - \left(-\frac{\sqrt{3}}{3}\right)\left(-\frac{\sqrt{5}}{6}\right) = \frac{\sqrt{62}}{18} - \frac{\sqrt{15}}{18} = \frac{\sqrt{186} - \sqrt{15}}{18}.$$

Find the Exact Value of the Expression Using the Provided cos⁡(s+t)\cos ( \mathrm { s } + \mathrm { t } )cos(s+t)

Find the Exact Value of the Expression Using the Provided $\cos ( \mathrm { s } + \mathrm { t } )$