Find the diagonal $ d_2 $ of a rhombus if side $a = 5$ and angle $\alpha = 37^o$.

$d_2 = 6.7438$

STEP 1: find angle $ \beta $

To find angle $ \beta $ use formula:

$$ \alpha + \beta = 90^o $$

After substituting $ \alpha = 37^o $ we have:

$$ 37^o + \beta = 90^o $$ $$ \beta = 90^o - 37^o $$ $$ \beta = 53^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ a } $$

After substituting $ \alpha = 37^o $ and $ a = 5 $ we have:

$$ \sin( 37^o ) = \dfrac{ h }{ 5 } $$ $$ 0.6018 = \dfrac{ h }{ 5 } $$$$ h = 0.6018 \cdot 5 $$$$ h = 3.0091 $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = \frac{ 53 }{ 2 }^o $ and $ h = 3.0091 $ we have:

$$ \sin \left( \frac{ 53^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( \frac{ 53 }{ 2 }^o ) = \dfrac{ 3.0091 }{ d_2 } $$ $$ 0.4462 = \dfrac{ 3.0091 }{ d_2 } $$ $$ d_2 = \dfrac{ 3.0091 }{ 0.4462 } $$ $$ d_2 = 6.7438 $$

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