STEP 1: find angle $ \beta $
To find angle $ \beta $ use Thw Law of Sines:
$$ \dfrac{ \sin( \beta )} { b } = \dfrac{ \sin( \gamma )} { c } $$After substituting $b = 31$, $c = 32$ and $\gamma = 48^o$ we have:
$$ \dfrac{ \sin( \beta )} { 31 } = \dfrac{ \sin( 48^o )} { 32 } $$ $$ \dfrac{ \sin( \beta )} { 31 } = \dfrac{ 0.7431 } { 32 } $$ $$ \sin( \beta ) \cdot 32 = 31 \cdot 0.7431 $$ $$ \sin( \beta ) \cdot 32 = 23.0375 $$ $$ \sin( \beta ) = \dfrac{ 23.0375 }{ 32 } $$ $$ \sin( \beta ) = 0.7199 $$ $$ \beta = \arcsin{ \left( 0.7199 \right)} $$ $$ \beta \approx 46.048^o $$STEP 2: find angle $ \alpha $
To find angle $ \alpha $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \beta = 46.048^o $ and $ \gamma = 48^o $ we have:
$$ \alpha + 46.048^o + 48^o = 180^o $$ $$ \alpha + 94.048^o = 180^o $$ $$ \alpha = 180^o - 94.048^o $$ $$ \alpha = 85.952^o $$STEP 3: 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 $b = 31$, $c = 32$ and $\alpha = 85.952^o$ we have:
$$ a^2 = 31^2 + 32^2 - 2 \cdot 31 \cdot 32 \cdot \cos( 85.952^o ) $$ $$ a^2 = 961 + 1024 - 2 \cdot 31 \cdot 32 \cdot \cos( 85.952^o ) $$ $$ a^2 = 1985 - 2 \cdot 992 \cdot \cos( 85.952^o ) $$ $$ a^2 = 1985 - 1984 \cdot 0.0706 $$ $$ a^2 = 1985 - 140.055 $$ $$ a^2 = 1844.945 $$ $$ a = \sqrt{ 1844.945 } $$$$ a \approx 42.9528 $$