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{1}{7} \\) and \\( \\sin t=-\\frac{4}{7}, s \\) in quadrant ii and \\( t \\) in quadrant iv\n(a) \\( \\sin (s+t)= \\) \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

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{1}{7} \\) and \\( \\sin t=-\\frac{4}{7}, s \\) in quadrant ii and \\( t \\) in quadrant iv\n(a) \\( \\sin (s+t)= \\) \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find (\cos s) and (\cos t)

Using the identity (\sin^{2}\theta+\cos^{2}\theta = 1), for (s) (quadrant II, (\cos s<0)): (\cos s=-\sqrt{1-\sin^{2}s}=-\sqrt{1 - (\frac{1}{7})^{2}}=-\sqrt{\frac{49 - 1}{49}}=-\frac{4\sqrt{3}}{7}) For (t) (quadrant IV, (\cos t>0)): (\cos t=\sqrt{1-\sin^{2}t}=\sqrt{1-(-\frac{4}{7})^{2}}=\sqrt{\frac{49 - 16}{49}}=\frac{\sqrt{33}}{7})

Step2: Use the sum formula for (\sin(s + t))

The sum formula (\sin(A + B)=\sin A\cos B+\cos A\sin B) Here (A = s), (B=t) (\sin(s + t)=\sin s\cos t+\cos s\sin t) Substitute (\sin s=\frac{1}{7}), (\cos s=-\frac{4\sqrt{3}}{7}), (\sin t=-\frac{4}{7}), (\cos t=\frac{\sqrt{33}}{7}) (\sin(s + t)=\frac{1}{7}\times\frac{\sqrt{33}}{7}+(-\frac{4\sqrt{3}}{7})\times(-\frac{4}{7})) (=\frac{\sqrt{33}}{49}+\frac{16\sqrt{3}}{49}=\frac{\sqrt{33}+16\sqrt{3}}{49})

Answer:

(\frac{\sqrt{33}+16\sqrt{3}}{49})