Find the diagonal $ d_1 $ of a rhombus if diagonal $d_2 = 9$ and angle $\beta = 54^o$.

$d_1 = 29.1246$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ \beta = 54^o $ we have:

$$ \alpha + 54^o = 90^o $$ $$ \alpha = 90^o - 54^o $$ $$ \alpha = 36^o $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ \alpha = 18^o $ and $ d_2 = 9 $ we have:

$$ \sin \left( \frac{ 36^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( 18^o ) = \dfrac{ 9 }{ d_1 } $$ $$ 0.309 = \dfrac{ 9 }{ d_1 } $$ $$ d_1 = \dfrac{ 9 }{ 0.309 } $$ $$ d_1 = 29.1246 $$

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