ابتدا با توجه به $\sin \alpha =\frac{5}{13}$، مقدار $\cos \alpha $ را بدست میآوریم:
${{\cos }^{2}}\alpha =1-{{\sin }^{2}}\alpha =1-{{\left( \frac{5}{13} \right)}^{2}}=1-\frac{25}{169}=\frac{144}{169}$
$\xrightarrow{\alpha }\cos \alpha =\frac{12}{13}$
با توجه به $\tan \beta =\frac{3}{4}$، مقدار $\sin \beta $ و $\cos \beta $ را بدست میآوریم.
$1+{{\tan }^{2}}\beta =\frac{1}{{{\cos }^{2}}\beta }\Rightarrow 1+\frac{9}{16}=\frac{1}{{{\cos }^{2}}\beta }$
$\Rightarrow \frac{25}{16}=\frac{1}{{{\cos }^{2}}\beta }\xrightarrow{\beta }\cos \beta =\frac{4}{5}$
${{\sin }^{2}}\beta =1-{{\cos }^{2}}\beta =1-{{\left( \frac{4}{5} \right)}^{2}}=1-\frac{16}{25}=\frac{9}{25}$
$\xrightarrow{\beta }\sin \beta =\frac{3}{5}$
بنابراین $\sin (\alpha +\beta )$ برابر است با:
$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta $
$\Rightarrow \sin (\alpha +\beta )=\frac{5}{13}\left( \frac{4}{5} \right)+\left( \frac{12}{13} \right)\left( \frac{3}{5} \right)=\frac{56}{65}$
