STEP 1: find angle $ \beta $
To find angle $ \beta $ use Thw Law of Sines:
$$ \dfrac{ \sin( \beta )} { b } = \dfrac{ \sin( \gamma )} { c } $$After substituting $b = 20$, $c = 25$ and $\gamma = 93^o$ we have:
$$ \dfrac{ \sin( \beta )} { 20 } = \dfrac{ \sin( 93^o )} { 25 } $$ $$ \dfrac{ \sin( \beta )} { 20 } = \dfrac{ 0.9986 } { 25 } $$ $$ \sin( \beta ) \cdot 25 = 20 \cdot 0.9986 $$ $$ \sin( \beta ) \cdot 25 = 19.9726 $$ $$ \sin( \beta ) = \dfrac{ 19.9726 }{ 25 } $$ $$ \sin( \beta ) = 0.7989 $$ $$ \beta = \arcsin{ \left( 0.7989 \right)} $$ $$ \beta \approx 53.0255^o $$STEP 2: find angle $ \alpha $
To find angle $ \alpha $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \beta = 53.0255^o $ and $ \gamma = 93^o $ we have:
$$ \alpha + 53.0255^o + 93^o = 180^o $$ $$ \alpha + 146.0255^o = 180^o $$ $$ \alpha = 180^o - 146.0255^o $$ $$ \alpha = 33.9745^o $$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 $b = 20$, $c = 25$ and $\alpha = 33.9745^o$ we have:
$$ a^2 = 20^2 + 25^2 - 2 \cdot 20 \cdot 25 \cdot \cos( 33.9745^o ) $$ $$ a^2 = 400 + 625 - 2 \cdot 20 \cdot 25 \cdot \cos( 33.9745^o ) $$ $$ a^2 = 1025 - 2 \cdot 500 \cdot \cos( 33.9745^o ) $$ $$ a^2 = 1025 - 1000 \cdot 0.8293 $$ $$ a^2 = 1025 - 829.2867 $$ $$ a^2 = 195.7133 $$ $$ a = \sqrt{ 195.7133 } $$$$ a \approx 13.9898 $$