STEP 1: find angle $ \gamma $
To find angle $ \gamma $ use Thw Law of Sines:
$$ \dfrac{ \sin( \alpha )} { a } = \dfrac{ \sin( \gamma )} { c } $$After substituting $a = 24$, $\alpha = 127^o$ and $c = 18$ we have:
$$ \dfrac{ \sin( 127^o )} { 24 } = \dfrac{ \sin( \gamma )} { 18 } $$ $$ \dfrac{ 0.7986 } { 24 } = \dfrac{ \sin( \gamma ) } { 18 } $$ $$ \sin( \gamma ) \cdot 24 = 0.7986 \cdot 18 $$ $$ \sin( \gamma ) \cdot 24 = 14.3754 $$ $$ \sin( \gamma ) = \dfrac{ 14.3754 }{ 24 } $$ $$ \sin( \gamma ) = 0.599 $$ $$ \gamma = \arcsin{ 0.599 } $$ $$ \gamma \approx 36.7966^o $$STEP 2: find angle $ \beta $
To find angle $ \beta $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \alpha = 127^o $ and $ \gamma = 36.7966^o $ we have:
$$ 127^o + \beta + 36.7966^o = 180^o $$ $$ \beta + 163.7966^o = 180^o $$ $$ \beta = 180^o - 163.7966^o $$ $$ \beta = 16.2034^o $$STEP 3: find side $ b $
To find side $ b $ use Law of Cosines:
$$ b^2 = a^2 + c^2 - 2 \cdot a \cdot c \cdot \cos( \beta ) $$After substituting $a = 24$, $c = 18$ and $\beta = 16.2034^o$ we have:
$$ b^2 = 24^2 + 18^2 - 2 \cdot 24 \cdot 18 \cdot \cos( 16.2034^o ) $$ $$ b^2 = 576 + 324 - 2 \cdot 24 \cdot 18 \cdot \cos( 16.2034^o ) $$ $$ b^2 = 900 - 2 \cdot 432 \cdot \cos( 16.2034^o ) $$ $$ b^2 = 900 - 864 \cdot 0.9603 $$ $$ b^2 = 900 - 829.6796 $$ $$ b^2 = 70.3204 $$ $$ b = \sqrt{ 70.3204 } $$$$ b \approx 8.3857 $$