Find the angle $ \alpha $ of a rhombus if side $a = 8$ and diagonal $d_1 = \frac{ 113 }{ 10 }$.

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{ 113 }{ 10 } $ and $ a = 8 $ we have:

$$ \left(\frac{ 113 }{ 10 }\right)^2 + d_2^2 = 4 \cdot 8^2 $$ $$ \frac{ 12769 }{ 100 } + d_2^2 = 256 $$ $$ d_2^2 = 256 - \frac{ 12769 }{ 100 } $$ $$ d_2^2 = \frac{ 12831 }{ 100 } $$ $$ d_2 = \sqrt{ \frac{ 12831 }{ 100 } } $$$$ d_2 = \frac{\sqrt{ 12831 }}{ 10 } $$

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{ 12831 }}{ 10 } $ and $ d_1 = \frac{ 113 }{ 10 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{\sqrt{ 12831 }}{ 10 } }{ \frac{ 113 }{ 10 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 12831 }}{ 113 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 12831 }}{ 113 } \right) $$

$ \arcsin(1.002) $ is not defined $ \Longrightarrow $ The problem has no solution.

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