use the given information to find (a) \\( \\sin (s+t) \\), (b) \\( \\tan (s+t) \\), and (c) the quadrant of…

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

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

Answer

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