Find the angle $ \beta $ of a rhombus if diagonal $d_1 = 40$ and diagonal $d_2 = 15$.

$\beta = 45.9514^o$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ d_2 = 15 $ and $ d_1 = 40 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 15 }{ 40 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 8 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 8 } \right) $$ $$ \frac{ \alpha }{ 2 } = 22.0243^o $$$$ \alpha = 22.0243^o \cdot 2 $$$$ \alpha = 44.0486^o $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 44.0486^o $ we have:

$$ 44.0486^o + \beta = 90^o $$ $$ \beta = 90^o - 44.0486^o $$ $$ \beta = 45.9514^o $$

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