Question

if 8tanA=-15 and 25 sinB= -7 and neither A and nor B is in 4quadrant then show that sinA cosB+cosAsinB=-304/425​

Answer 1

Answer:

your solution is as follows

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Step-by-step explanation:

$to \: prove \: that : \\ sin \: A.co s\: B + cos \: A.sin \: B = - \frac{ 304}{425} \\ \\ given : \\ tan \: A = - \frac{8}{15} \\ \\ sin \: B = - \frac{7}{25} \\ \\ since \: we \: know \: that \\ tangent \: angle \: angle \: is \: always \\ negative \: in \: quadrant \: 2 \: and \: 4 \\ but \: as \: angle \: A \: doesnt \: lie \: in\\ quadrant \: 4 \: \\ angle \: A \: must \: lie \: in \: quadrant \: 2 \\ \\ similarly \: we \: know \: that \\ sine \: angle \: is \: always \: negative \: in \\ quadrant \: 3 \: and \:quadrant \: 4 \\ but \: since \: angle \: B \: is \: not \: lying \\ in \: quadrant \: 4 \\ angle \: B \: must \: lie \: in \: quadrant \: 3$

$so \: now \: using \\ 1 + tan {}^{2} \: A = sec {}^{2} \: A \\ \\ = 1 + ( - \frac{15}{8} \: ) {}^{2} \\ \\ = 1 + \frac{225}{64} \\ \\ = \frac{225 + 64}{64} \\ \\ = \frac{289}{64} \\ \\ sec \: A = \frac{17}{8} \\ \\ cos \: A = \frac{8}{17} \\ \\ since \: angle \:A \:lies \: in \: quadrant \: 2 \\ cos \: A = - \frac{8}{17} \\ \\ similarly \: by \: applying \\ sin {}^{2} \: A + cos {}^{2} \: A = 1 \\ we \: get \\ \\ sin \: A = \frac{15}{17} \\ \\ since \: angle \: A\: lies \: in \: quadrant \: A \\ sin \: A = \frac{15}{17}$

$accordingly \: here \\ sin \: B = - \frac{7}{25} \\ \\ using \: now \\ sin {}^{2} \: B + cos {}^{2} \: B = 1 \\ cos {}^{2} \: B = 1 - sin {}^{2} \: B \\ \\ = 1 - ( - \frac{7}{25} \: ) {}^{2} \\ \\ = 1 - \frac{49}{625} \\ \\ = \frac{625 - 49}{625} \\ \\ = \frac{576}{625} \\ \\ cos \: B = \frac{24}{25} \\ \\ but \: as \: angle \: B \: lies \: in \: quadrant \: 3 \\ cos \: B = - \frac{24}{25}$

$now \: considering \: \\ sin \: A. cos \: B+cos \: .Asin \: B=- \frac{304}{425} \\ \\ LHS = sin \: A .cos \: B+cos \: A.sin \: B \\ \\ = \frac{15}{17} \: \times ( - \frac{24}{25} \: ) + ( - \frac{8}{17} ) \times ( - \frac{7}{25} ) \\ \\ = \frac{15( - 24) +( - 7)( - 8) }{17 \times 25} \\ \\ = \frac{ - 360 + 56}{425} \\ \\ = - \frac{304}{425} \\ \\ = RHS$