Find the angle $ \beta $ of a rhombus if diagonal $d_1 = \frac{ 1256 }{ 5 }$ and diagonal $d_2 = \frac{ 2497 }{ 10 }$.

Problem has no solutions.

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 = \frac{ 2497 }{ 10 } $ and $ d_1 = \frac{ 1256 }{ 5 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 2497 }{ 10 } }{ \frac{ 1256 }{ 5 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 2497 }{ 2512 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 2497 }{ 2512 } \right) $$ $$ \frac{ \alpha }{ 2 } = 83.7355^o $$$$ \alpha = 83.7355^o \cdot 2 $$$$ \alpha = 167.4709^o $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 167.4709^o $ we have:

$$ 167.4709^o + \beta = 90^o $$ $$ \beta = 90^o - 167.4709^o $$ $$ \beta = -77.4709^o $$

The result has to be greater than zero. $ \Longrightarrow $ The problem has no solution.

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