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 $a = 535$, $b = 340$ and $\gamma = 71^o$ we have:
$$ c^2 = 535^2 + 340^2 - 2 \cdot 535 \cdot 340 \cdot \cos( 71^o ) $$ $$ c^2 = 286225 + 115600 - 2 \cdot 535 \cdot 340 \cdot \cos( 71^o ) $$ $$ c^2 = 401825 - 2 \cdot 181900 \cdot \cos( 71^o ) $$ $$ c^2 = 401825 - 363800 \cdot 0.3256 $$ $$ c^2 = 401825 - 118441.6946 $$ $$ c^2 = 283383.3054 $$ $$ c = \sqrt{ 283383.3054 } $$$$ c \approx 532.3376 $$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:
$$ 535^2 = 340^2 + 532.3376^2 - 2 \cdot 340 \cdot 532.3376 \cdot \cos( \alpha ) $$ $$ 286225 = 115600 + 283383.3054 - 361989.5584 \cos( \alpha ) $$ $$ 361989.5584 \cos( \alpha ) = 115600 + 283383.3054 - 286225 $$ $$ 361989.5584 \cos( \alpha ) = 112758.3054 $$ $$ 361989.5584 \cos( \alpha ) = 112758.3054 $$ $$ \cos( \alpha ) = 0.3115 $$ $$ \alpha = \arccos{ \left( 0.3115 \right)} $$ $$ \alpha \approx 71.8506^o $$