Find the diagonal $ d_2 $ of a rhombus if side $a = 4$ and angle $\alpha = 30^o$.

$d_2 = 4$

STEP 1: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 30^o $ we have:

$$ 30^o + \beta = 90^o $$ $$ \beta = 90^o - 30^o $$ $$ \beta = 60^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 30^o $ and $ a = 4 $ we have:

$$ \sin( 30^o ) = \dfrac{ h }{ 4 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ h }{ 4 } $$$$ h = \frac{ 1 }{ 2 } \cdot 4 $$$$ h = 2 $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ \beta = 30^o $ and $ h = 2 $ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 30^o ) = \dfrac{ 2 }{ d_2 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ 2 }{ d_2 } $$ $$ d_2 = \dfrac{ 2 }{ \frac{ 1 }{ 2 } } $$ $$ d_2 = 4 $$

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