Question

if a and b are acute angles such that sinA=4/5 and cosB=12/13, calculate, without using tables: sin(a-b)

Answer 1

$\sin \alpha = \frac{4}{5} \\ \cos \alpha = \sqrt{1-\sin^2 \alpha }= \sqrt{1- \frac{16}{25} } = \sqrt{ \frac{9}{25} }= \frac{3}{5} \\ \\ \cos \beta = \frac{12}{13} \\ \sin \beta = \sqrt{1-\cos^2 \beta }= \sqrt{1- \frac{144}{169} } = \sqrt{ \frac{25}{169} }= \frac{5}{13}$


$\\ \\ \sin (\alpha- \beta ) =\sin \alpha \cos \beta -\cos \alpha \sin \beta = \frac{4}{5}\times \frac{12}{13}- \frac{3}{5}\times \frac{5}{13} = \frac{48}{65}- \frac{15}{65}= \frac{33}{65}$

Answer:
sin(α-β) = 33/65