Find the angle $ \beta $ of a rhombus if side $a = 13$ and diagonal $d_1 = 12$.

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 = 12 $ and $ a = 13 $ we have:

$$ 12^2 + d_2^2 = 4 \cdot 13^2 $$ $$ 144 + d_2^2 = 676 $$ $$ d_2^2 = 676 - 144 $$ $$ d_2^2 = 532 $$ $$ d_2 = \sqrt{ 532 } $$$$ d_2 = 2 \sqrt{ 133 } $$

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 = 2 \sqrt{ 133 } $ and $ d_1 = 12 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 2 \sqrt{ 133 } }{ 12 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 133 }}{ 6 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 133 }}{ 6 } \right) $$

$ \arcsin(1.922) $ is not defined $ \Longrightarrow $ The problem has no solution.

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