Cho \(\sin \alpha = \frac{3}{5},\cos \beta = \frac{{12}}{{13}}\) và \({0^0}
Sử dụng kiến thức về công thức cộng để tính: \(\sin \left( {\alpha + \beta } \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \), \(\cos \left( {\alpha - \beta } \right) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).
Vì \({0^0} 0,\sin \beta > 0\)
Do đó, \(\cos \alpha \) \( = \sqrt {1 - {{\sin }^2}\alpha } \) \( = \sqrt {1 - {{\left( {\frac{3}{5}} \right)}^2}} \) \( = \frac{4}{5}\),\(\sin \beta \) \( = \sqrt {1 - {{\cos }^2}\beta } \) \( = \sqrt {1 - {{\left( {\frac{{12}}{{13}}} \right)}^2}} \) \( = \frac{5}{{13}}\)
\(\sin \left( {\alpha + \beta } \right) \) \( = \sin \alpha \cos \beta + \cos \alpha \sin \beta \) \( = \frac{3}{5}.\frac{{12}}{{13}} + \frac{4}{5}.\frac{5}{{13}} \) \( = \frac{{56}}{{65}}\)
\(\cos \left( {\alpha - \beta } \right) \) \( = \cos \alpha \cos \beta + \sin \alpha \sin \beta \) \( = \frac{4}{5}.\frac{{12}}{{13}} + \frac{3}{5}.\frac{5}{{13}} \) \( = \frac{{63}}{{65}}\)