Find the angle $ \alpha $ of a rhombus if side $a = \frac{ 37 }{ 2 }$ and diagonal $d_1 = \frac{ 126 }{ 5 }$.

Problem has no solutions.

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ d_1 = \frac{ 126 }{ 5 } $ and $ a = \frac{ 37 }{ 2 } $ we have:

$$ \left(\frac{ 126 }{ 5 }\right)^2 + d_2^2 = 4 \cdot \left(\frac{ 37 }{ 2 }\right)^2 $$ $$ \frac{ 15876 }{ 25 } + d_2^2 = 1369 $$ $$ d_2^2 = 1369 - \frac{ 15876 }{ 25 } $$ $$ d_2^2 = \frac{ 18349 }{ 25 } $$ $$ d_2 = \sqrt{ \frac{ 18349 }{ 25 } } $$$$ d_2 = \frac{\sqrt{ 18349 }}{ 5 } $$

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 = \frac{\sqrt{ 18349 }}{ 5 } $ and $ d_1 = \frac{ 126 }{ 5 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{\sqrt{ 18349 }}{ 5 } }{ \frac{ 126 }{ 5 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 18349 }}{ 126 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 18349 }}{ 126 } \right) $$

$ \arcsin(1.075) $ is not defined $ \Longrightarrow $ The problem has no solution.

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