Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin s = \frac { 1 } { 4 }$, with $s$ in quadrant II, and $\sin t = - \frac { 1 } { 2 }$, with $t$ in quadrant IV.
A) $\frac { 1 + 3 \sqrt { 5 } } { 8 }$
B) $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$
C) $\frac { 1 - 3 \sqrt { 5 } } { 8 }$
D) $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$

Question

Quizplus · Accepted Answer

To find $\tan(s + t)$, we use the identity $\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}$. First, we find $\tan s$ and $\tan t$.For $s$ in quadrant II, where $\sin s = \frac{1}{4}$, we use the Pythagorean identity to find $\cos s$. Since $\sin^2 s + \cos^2 s = 1$, we have $\cos s = \sqrt{1 - \sin^2 s} = \sqrt{1 - \left(\frac{1}{4}\right)^2} = \sqrt{\frac{15}{16}}$. In quadrant II, $\cos s$ is negative, so $\cos s = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4}$. Thus, $\tan s = \frac{\sin s}{\cos s} = \frac{\frac{1}{4}}{-\frac{\sqrt{15}}{4}} = -\frac{1}{\sqrt{15}}$.For $t$ in quadrant IV, where $\sin t = -\frac{1}{2}$, we find $\cos t$ similarly. $\cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4}}$. In quadrant IV, $\cos t$ is positive, so $\cos t = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$. Thus, $\tan t = \frac{\sin t}{\cos t} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$.Now, $\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} = \frac{-\frac{1}{\sqrt{15}} - \frac{1}{\sqrt{3}}}{1 - (-\frac{1}{\sqrt{15}})(-\frac{1}{\sqrt{3}})} = \frac{-\frac{\sqrt{3}}{\sqrt{15}} - \frac{\sqrt{15}}{\sqrt{15}}}{1 - \frac{1}{\sqrt{45}}} = \frac{-\frac{\sqrt{3} + \sqrt{15}}{\sqrt{15}}}{\frac{\sqrt{45} - 1}{\sqrt{45}}}$.Simplifying further, we get $\tan(s + t) = \frac{-\sqrt{3} - \sqrt{15}}{3 - \sqrt{15}}$, which after rationalizing the denominator and simplifying, matches the form of option B: $\frac{4\sqrt{3} + \sqrt{15}}{-11}$.

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

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