Find the angle $ \beta $ of a rhombus if diagonal $d_1 = 15$ and diagonal $d_2 = \frac{ 66 }{ 5 }$.

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{ 66 }{ 5 } $ and $ d_1 = 15 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 66 }{ 5 } }{ 15 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 22 }{ 25 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 22 }{ 25 } \right) $$ $$ \frac{ \alpha }{ 2 } = 61.6424^o $$$$ \alpha = 61.6424^o \cdot 2 $$$$ \alpha = 123.2847^o $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 123.2847^o $ we have:

$$ 123.2847^o + \beta = 90^o $$ $$ \beta = 90^o - 123.2847^o $$ $$ \beta = -33.2847^o $$

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

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