Find the diagonal $ d_1 $ of a rhombus if side $a = 105$ and angle $\alpha = 55^o$.

$d_1 = 186.2723$

STEP 1: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 55^o $ and $ a = 105 $ we have:

$$ \sin( 55^o ) = \dfrac{ h }{ 105 } $$ $$ 0.8192 = \dfrac{ h }{ 105 } $$$$ h = 0.8192 \cdot 105 $$$$ h = 86.011 $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = \frac{ 55 }{ 2 }^o $ and $ h = 86.011 $ we have:

$$ \sin \left( \frac{ 55^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( \frac{ 55 }{ 2 }^o ) = \dfrac{ 86.011 }{ d_1 } $$ $$ 0.4617 = \dfrac{ 86.011 }{ d_1 } $$ $$ d_1 = \dfrac{ 86.011 }{ 0.4617 } $$ $$ d_1 = 186.2723 $$

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