Find the angle $ \alpha $ of a rhombus if side $a = \frac{ 33 }{ 8 }$ and diagonal $d_2 = \frac{ 127 }{ 16 }$.

Problem has no solutions.

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 = \frac{ 127 }{ 16 } $ and $ a = \frac{ 33 }{ 8 } $ we have:

$$ d_1 ^ {\,2} + \left(\frac{ 127 }{ 16 }\right)^2 = 4 \cdot \left(\frac{ 33 }{ 8 }\right)^2 $$ $$ d_1 ^ {\,2} + \frac{ 16129 }{ 256 } = \frac{ 1089 }{ 16 } $$ $$ d_1 ^ {\,2} = \frac{ 1089 }{ 16 } - \frac{ 16129 }{ 256 } $$ $$ d_1 ^ {\,2} = \frac{ 1295 }{ 256 } $$ $$ d_1 = \sqrt{ \frac{ 1295 }{ 256 } } $$$$ d_1 = \frac{\sqrt{ 1295 }}{ 16 } $$

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{ 127 }{ 16 } $ and $ d_1 = \frac{\sqrt{ 1295 }}{ 16 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 127 }{ 16 } }{ \frac{\sqrt{ 1295 }}{ 16 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 127 \sqrt{ 1295}}{ 1295 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 127 \sqrt{ 1295}}{ 1295 } \right) $$

$ \arcsin(3.529) $ is not defined $ \Longrightarrow $ The problem has no solution.

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