# How do you find the exact value of sin(alpha-beta) if sinalpha=-5/13 and sinbeta=1/6 if the terminal side of alpha lies in QIII and the terminal side of beta lies in QI?

Jun 20, 2017

$\frac{12 - 5 \sqrt{35}}{78} = 0.225$

#### Explanation:

$\sin \alpha = - \frac{5}{13}$ since $\alpha$ lie in QIII, therefore $\cos \alpha = - \frac{12}{13} \mathmr{and} \tan \alpha = \frac{5}{12}$

$\sin \beta = \frac{1}{6}$ since $\beta$ lie in QI, therefore $\cos \beta = \frac{\sqrt{35}}{6} \mathmr{and} \tan \alpha = \frac{1}{\sqrt{35}}$

$\sin \left(\alpha - \beta\right) = \sin \alpha \cos \beta - \sin \beta \cos \alpha$ $\to i$

plug in the above values in $\to i$

$\sin \left(\alpha - \beta\right) = \left(- \frac{5}{13}\right) \left(\frac{\sqrt{35}}{6}\right) - \left(\frac{1}{6}\right) \left(- \frac{12}{13}\right)$

$\sin \left(\alpha - \beta\right) = \frac{- 5 \sqrt{35}}{78} + \frac{12}{78} = \frac{12 - 5 \sqrt{35}}{78} = 0.225$