Question

use the given information to find (a) \\( \sin (s+t) \\), (b) \\( \tan (s+t) \\), and (c) the quadrant of \\( s+t \\).
\\( \sin s=\frac{5}{7} \\) and \\( \sin t=-\frac{6}{7} \\), s in quadrant ii and t in quadrant iv
(a) \\( \sin (s+t)=\frac{5 \sqrt{13}+12 \sqrt{6}}{49} \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
(b) \\( \tan (s+t)=\frac{3 \sqrt{13}+5 \sqrt{6}}{6} \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
(c) what is the quadrant of \\( s+t \\)?
quadrant iii
quadrant i
quadrant ii
quadrant iv

Explanation:

Step1: Analyze the sign of \(\sin(s + t)\) and \(\tan(s + t)\)

We know that \(\sin(s + t)=\frac{5\sqrt{13}+ 12\sqrt{6}}{49}>0\) (since the numerator \(5\sqrt{13}+12\sqrt{6}>0\) as \(\sqrt{13}>0\) and \(\sqrt{6}>0\)) and \(\tan(s + t)=\frac{3\sqrt{13}+5\sqrt{6}}{6}>0\) (since the numerator \(3\sqrt{13}+5\sqrt{6}>0\) as \(\sqrt{13}>0\) and \(\sqrt{6}>0\)).

Step2: Determine the quadrant based on the signs of trigonometric functions

In Quadrant I, \(\sin\theta>0\) and \(\tan\theta>0\). In Quadrant II, \(\sin\theta>0\) and \(\tan\theta<0\). In Quadrant III, \(\sin\theta<0\) and \(\tan\theta>0\). In Quadrant IV, \(\sin\theta<0\) and \(\tan\theta<0\).

Since \(\sin(s + t)>0\) and \(\tan(s + t)>0\), the angle \(s + t\) must be in a quadrant where both sine and tangent are positive.

Answer:

Quadrant I