Find the angle $ \beta $ of a rhombus if side $a = 4$ and diagonal $d_2 = 4$.

$\beta = 19.4712^o$

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 = 4 $ and $ a = 4 $ we have:

$$ d_1 ^ {\,2} + 4^2 = 4 \cdot 4^2 $$ $$ d_1 ^ {\,2} + 16 = 64 $$ $$ d_1 ^ {\,2} = 64 - 16 $$ $$ d_1 ^ {\,2} = 48 $$ $$ d_1 = \sqrt{ 48 } $$$$ d_1 = 4 \sqrt{ 3 } $$

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 = 4 $ and $ d_1 = 4 \sqrt{ 3 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 4 }{ 4 \sqrt{ 3 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 3 }}{ 3 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 3 }}{ 3 } \right) $$ $$ \frac{ \alpha }{ 2 } = 35.2644^o $$$$ \alpha = 35.2644^o \cdot 2 $$$$ \alpha = 70.5288^o $$

STEP 3: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 70.5288^o $ we have:

$$ 70.5288^o + \beta = 90^o $$ $$ \beta = 90^o - 70.5288^o $$ $$ \beta = 19.4712^o $$

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