STEP 1: find side $ c $
To find side $ c $ use Law of Cosines:
$$ c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos( \gamma ) $$After substituting $a = \frac{ 557 }{ 100 }$, $b = \frac{ 103 }{ 25 }$ and $\gamma = \frac{ 137 }{ 5 }^o$ we have:
$$ c^2 = \left(\frac{ 557 }{ 100 }\right)^2 + \left(\frac{ 103 }{ 25 }\right)^2 - 2 \cdot \frac{ 557 }{ 100 } \cdot \frac{ 103 }{ 25 } \cdot \cos( \frac{ 137 }{ 5 }^o ) $$ $$ c^2 = \frac{ 310249 }{ 10000 } + \frac{ 10609 }{ 625 } - 2 \cdot \frac{ 557 }{ 100 } \cdot \frac{ 103 }{ 25 } \cdot \cos( \frac{ 137 }{ 5 }^o ) $$ $$ c^2 = \frac{ 479993 }{ 10000 } - 2 \cdot \frac{ 57371 }{ 2500 } \cdot \cos( \frac{ 137 }{ 5 }^o ) $$ $$ c^2 = \frac{ 479993 }{ 10000 } - \frac{ 57371 }{ 1250 } \cdot 0.8878 $$ $$ c^2 = \frac{ 479993 }{ 10000 } - 40.7479 $$ $$ c^2 = 7.2514 $$ $$ c = \sqrt{ 7.2514 } $$$$ c \approx 2.6928 $$STEP 2: find angle $ \alpha $
To find angle $ \alpha $ use Law of Cosines:
$$ a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos( \alpha ) $$After substituting we have:
$$ \left(\frac{ 557 }{ 100 }\right)^2 = \left(\frac{ 103 }{ 25 }\right)^2 + 2.6928^2 - 2 \cdot \frac{ 103 }{ 25 } \cdot 2.6928 \cdot \cos( \alpha ) $$ $$ \frac{ 310249 }{ 10000 } = \frac{ 10609 }{ 625 } + 7.2514 - 22.189 \cos( \alpha ) $$ $$ 22.189 \cos( \alpha ) = \frac{ 10609 }{ 625 } + 7.2514 - \frac{ 310249 }{ 10000 } $$ $$ 22.189 \cos( \alpha ) = -6.7991 $$ $$ 22.189 \cos( \alpha ) = -6.7991 $$ $$ \cos( \alpha ) = -0.3064 $$ $$ \alpha = \arccos{ \left( -0.3064 \right)} $$ $$ \alpha \approx 107.8434^o $$