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 $a = \frac{ 9 }{ 2 }$, $b = 8$ and $\gamma = 130^o$ we have:
$$ c^2 = \left(\frac{ 9 }{ 2 }\right)^2 + 8^2 - 2 \cdot \frac{ 9 }{ 2 } \cdot 8 \cdot \cos( 130^o ) $$ $$ c^2 = \frac{ 81 }{ 4 } + 64 - 2 \cdot \frac{ 9 }{ 2 } \cdot 8 \cdot \cos( 130^o ) $$ $$ c^2 = \frac{ 337 }{ 4 } - 2 \cdot 36 \cdot \cos( 130^o ) $$ $$ c^2 = \frac{ 337 }{ 4 } - 72 \cdot \left(-0.6428\right) $$ $$ c^2 = \frac{ 337 }{ 4 } - \left(-46.2807\right) $$ $$ c^2 = 130.5307 $$ $$ c = \sqrt{ 130.5307 } $$$$ c \approx 11.425 $$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:
$$ \left(\frac{ 9 }{ 2 }\right)^2 = 8^2 + 11.425^2 - 2 \cdot 8 \cdot 11.425 \cdot \cos( \alpha ) $$ $$ \frac{ 81 }{ 4 } = 64 + 130.5307 - 182.8001 \cos( \alpha ) $$ $$ 182.8001 \cos( \alpha ) = 64 + 130.5307 - \frac{ 81 }{ 4 } $$ $$ 182.8001 \cos( \alpha ) = 174.2807 $$ $$ 182.8001 \cos( \alpha ) = 174.2807 $$ $$ \cos( \alpha ) = 0.9534 $$ $$ \alpha = \arccos{ \left( 0.9534 \right)} $$ $$ \alpha \approx 17.5612^o $$