Find the side $ a $ of a rhombus if diagonal $d_2 = 4$ and angle $\alpha = 70^o$.

$a = 4.0198$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 35^o $ and $ d_2 = 4 $ we have:

$$ \sin \left( \frac{ 70^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( 35^o ) = \dfrac{ 4 }{ d_1 } $$ $$ 0.5736 = \dfrac{ 4 }{ d_1 } $$ $$ d_1 = \dfrac{ 4 }{ 0.5736 } $$ $$ d_1 = 6.9738 $$

STEP 2: find side $ a $

To find side $ a $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_1 = 6.9738 $ and $ d_2 = 4 $ we have:

$$ 6.9738^2 + 4^2 = 4 \cdot a^2 $$ $$ 48.6337 + 16 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 64.6337 $$ $$ a^2 = \frac{ 64.6337 }{ 4 } $$ $$ a^2 = 16.1584 $$ $$ a = \sqrt{ 16.1584 } $$$$ a = 4.0198 $$

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