Find the angle $ \alpha $ of a rhombus if side $a = \frac{ 3 }{ 2 }$ and diagonal $d_2 = 2$.

$\alpha = 126.8699^o$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ d_2 = 2 $ and $ a = \frac{ 3 }{ 2 } $ we have:

$$ d_1 ^ {\,2} + 2^2 = 4 \cdot \left(\frac{ 3 }{ 2 }\right)^2 $$ $$ d_1 ^ {\,2} + 4 = 9 $$ $$ d_1 ^ {\,2} = 9 - 4 $$ $$ d_1 ^ {\,2} = 5 $$ $$ d_1 = \sqrt{ 5 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ d_2 = 2 $ and $ d_1 = \sqrt{ 5 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 2 }{ \sqrt{ 5 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 2 \sqrt{ 5}}{ 5 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 2 \sqrt{ 5}}{ 5 } \right) $$ $$ \frac{ \alpha }{ 2 } = 63.4349^o $$$$ \alpha = 63.4349^o \cdot 2 $$$$ \alpha = 126.8699^o $$

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