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

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) = \\square (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. sin s = \\frac{5}{7} and sin t = - \\frac{6}{7}, s in quadrant ii and t in quadrant iv (a) sin (s + t) = \\square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find (\cos s)

Since (\sin s=\frac{5}{7}) and (s) is in quadrant II. Using (\sin^{2}\theta+\cos^{2}\theta = 1), we have (\cos^{2}s=1-\sin^{2}s=1 - (\frac{5}{7})^{2}=1-\frac{25}{49}=\frac{24}{49}). So (\cos s=-\frac{2\sqrt{6}}{7}) (negative in quadrant II).

Step2: Find (\cos t)

Since (\sin t =-\frac{6}{7}) and (t) is in quadrant IV. Using (\sin^{2}\theta+\cos^{2}\theta = 1), we have (\cos^{2}t=1-\sin^{2}t=1-(-\frac{6}{7})^{2}=1-\frac{36}{49}=\frac{13}{49}). So (\cos t=\frac{\sqrt{13}}{7}) (positive in quadrant IV).

Step3: Use the sum formula for sine

The sum formula for (\sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = s) and (B=t). (\sin(s + t)=\sin s\cos t+\cos s\sin t) Substitute (\sin s=\frac{5}{7}), (\cos s=-\frac{2\sqrt{6}}{7}), (\sin t=-\frac{6}{7}), (\cos t=\frac{\sqrt{13}}{7}) [ \begin{align*} \sin(s + t)&=\frac{5}{7}\times\frac{\sqrt{13}}{7}+(-\frac{2\sqrt{6}}{7})\times(-\frac{6}{7})\ &=\frac{5\sqrt{13}+12\sqrt{6}}{49} \end{align*} ]

Answer:

(\frac{5\sqrt{13}+12\sqrt{6}}{49})