STEP 1: find angle $ \beta $
To find angle $ \beta $ use Thw Law of Sines:
$$ \dfrac{ \sin( \alpha )} { a } = \dfrac{ \sin( \beta )} { b } $$After substituting $a = 8$, $\alpha = 35^o$ and $b = 11$ we have:
$$ \dfrac{ \sin( 35^o )} { 8 } = \dfrac{ \sin( \beta )} { 11 } $$ $$ \dfrac{ 0.5736 } { 8 } = \dfrac{ \sin( \beta ) } { 11 } $$ $$ \sin( \beta ) \cdot 8 = 0.5736 \cdot 11 $$ $$ \sin( \beta ) \cdot 8 = 6.3093 $$ $$ \sin( \beta ) = \dfrac{ 6.3093 }{ 8 } $$ $$ \sin( \beta ) = 0.7887 $$ $$ \beta = \arcsin{ 0.7887 } $$ $$ \beta \approx 52.0612^o $$STEP 2: find angle $ \gamma $
To find angle $ \gamma $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \alpha = 35^o $ and $ \beta = 52.0612^o $ we have:
$$ 35^o + 52.0612^o + \gamma = 180^o $$ $$ \gamma + 87.0612^o = 180^o $$ $$ \gamma = 180^o - 87.0612^o $$ $$ \gamma = 92.9388^o $$STEP 3: 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 $a = 8$, $b = 11$ and $\gamma = 92.9388^o$ we have:
$$ c^2 = 8^2 + 11^2 - 2 \cdot 8 \cdot 11 \cdot \cos( 92.9388^o ) $$ $$ c^2 = 64 + 121 - 2 \cdot 8 \cdot 11 \cdot \cos( 92.9388^o ) $$ $$ c^2 = 185 - 2 \cdot 88 \cdot \cos( 92.9388^o ) $$ $$ c^2 = 185 - 176 \cdot \left(-0.0513\right) $$ $$ c^2 = 185 - \left(-9.0235\right) $$ $$ c^2 = 194.0235 $$ $$ c = \sqrt{ 194.0235 } $$$$ c \approx 13.9292 $$