STEP 1: find side $ a $
To find side $ a $ use Law of Cosines:
$$ a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos( \alpha ) $$After substituting $b = 15$, $c = 120$ and $\alpha = 58^o$ we have:
$$ a^2 = 15^2 + 120^2 - 2 \cdot 15 \cdot 120 \cdot \cos( 58^o ) $$ $$ a^2 = 225 + 14400 - 2 \cdot 15 \cdot 120 \cdot \cos( 58^o ) $$ $$ a^2 = 14625 - 2 \cdot 1800 \cdot \cos( 58^o ) $$ $$ a^2 = 14625 - 3600 \cdot 0.5299 $$ $$ a^2 = 14625 - 1907.7094 $$ $$ a^2 = 12717.2906 $$ $$ a = \sqrt{ 12717.2906 } $$$$ a \approx 112.771 $$STEP 2: find angle $ \beta $
To find angle $ \beta $ use Law of Cosines:
$$ b^2 = a^2 + c^2 - 2 \cdot a \cdot c \cdot \cos( \beta ) $$After substituting we have:
$$ 15^2 = 112.771^2 + 120^2 - 2 \cdot 112.771 \cdot 120 \cdot \cos( \beta ) $$ $$ 225 = 12717.2906 + 14400 - 27065.0317 \cos( \beta ) $$ $$ 27065.0317 \cos( \beta ) = 12717.2906 + 14400 - 225 $$ $$ 27065.0317 \cos( \beta ) = 26892.2906 $$ $$ 27065.0317 \cos( \beta ) = 26892.2906 $$ $$ \cos( \beta ) = 0.9936 $$ $$ \beta = \arccos{ \left( 0.9936 \right)} $$ $$ \beta \approx 6.4768^o $$