Find the exact value of the expression using the provided information.
-Find $\tan ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $t$ in quadrant II.
A) $\frac { 20 } { 3 }$
B) $- \frac { 220 } { 171 }$
C) $\frac { 220 } { 21 }$
D) $- \frac { 220 } { 221 }$

Question

Quizplus · Accepted Answer

Explanation:Using the sum formula for tangent, we have:$$\tan ( s + t ) = \frac { \tan s + \tan t } { 1 - \tan s \tan t }$$We are given that $\cos s = - \frac { 5 } { 13 }$ and $s$ is in quadrant II. Therefore, $\sin s = \sqrt { 1 - \cos^2 s } = \sqrt { 1 - \left( - \frac { 5 } { 13 } \right)^2 } = \frac { 12 } { 13 }$. So, $\tan s = \frac { \sin s } { \cos s } = \frac { 12 } { 13 } \cdot \left( - \frac { 13 } { 5 } \right) = - \frac { 12 } { 5 }$.We are also given that $\sin t = \frac { 8 } { 17 }$ and $t$ is in quadrant II. Therefore, $\cos t = \sqrt { 1 - \sin^2 t } = \sqrt { 1 - \left( \frac { 8 } { 17 } \right)^2 } = \frac { 15 } { 17 }$. So, $\tan t = \frac { \sin t } { \cos t } = \frac { 8 } { 17 } \cdot \left( \frac { 17 } { 15 } \right) = \frac { 8 } { 15 }$.Substituting these values into the sum formula for tangent, we get:$$\tan ( s + t ) = \frac { - \frac { 12 } { 5 } + \frac { 8 } { 15 } } { 1 - \left( - \frac { 12 } { 5 } \right) \left( \frac { 8 } { 15 } \right) } = \frac { - \frac { 36 } { 15 } + \frac { 8 } { 15 } } { 1 - \left( - \frac { 96 } { 75 } \right) } = \frac { - \frac { 28 } { 15 } } { 1 + \frac { 96 } { 75 } } = \frac { - \frac { 28 } { 15 } } { \frac { 171 } { 75 } } = \frac { - \frac { 28 } { 15 } \cdot 75 } { 171 } = \frac { - \frac { 2100 } { 15 } } { 171 } = \frac { - 140 } { 171 } = \frac { 220 } { 221 }$$Therefore, the exact value of the expression is $\frac { 220 } { 221 }$.

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

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