$a){\left|BC\right|}^2={\left|AB\right|}^2+{\left|AC\right|}^2-2\cdot \left|AB\right|\cdot \left|AC\right|\cdot \cos {120}^o$  

${\left|BC\right|}^2=144+36-2\cdot 12\cdot 6\cdot \cos \left({180}^o-{60}^o\right)$  

${\left|BC\right|}^2=180+144\cos {60}^o$  

${\left|BC\right|}^2=180+144\cdot \frac{1}{2}$  

${\left|BC\right|}^2=180+72$  

${\left|BC\right|}^2=252$  

$\left|BC\right|=\sqrt{252}=\sqrt{36}\cdot \sqrt{7}=6\sqrt{7}$  

 

Z twierdzenia sinusów:

$\frac{\left|BC\right|}{\sin \alpha }=\frac{\left|AC\right|}{\sin \beta }$  

$\sin \beta =\frac{\left|AC\right|}{\left|BC\right|}\cdot \sin \alpha \approx \frac{6}{6\cdot \sqrt{7}}\cdot \sin {60}^o\approx \frac{6}{6\cdot 2,6458}\cdot 0,866\approx 0,3273$   

Z tablic wartości funkcji trygonometrycznych:

$\sin {19}^o\approx 0,3256$  

$\beta \approx {19}^o$  

Trzeci kąt ma miarę:

$\gamma \approx {180}^o-\left({120}^o+{19}^o\right)={180}^o-{139}^o={41}^o$