Find the angle $ \beta $ of a rhombus if side $a = 12$ and diagonal $d_1 = 18$.

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 = 18 $ and $ a = 12 $ we have:

$$ 18^2 + d_2^2 = 4 \cdot 12^2 $$ $$ 324 + d_2^2 = 576 $$ $$ d_2^2 = 576 - 324 $$ $$ d_2^2 = 252 $$ $$ d_2 = \sqrt{ 252 } $$$$ d_2 = 6 \sqrt{ 7 } $$

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 = 6 \sqrt{ 7 } $ and $ d_1 = 18 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 6 \sqrt{ 7 } }{ 18 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 7 }}{ 3 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 7 }}{ 3 } \right) $$ $$ \frac{ \alpha }{ 2 } = 61.8745^o $$$$ \alpha = 61.8745^o \cdot 2 $$$$ \alpha = 123.749^o $$

STEP 3: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 123.749^o $ we have:

$$ 123.749^o + \beta = 90^o $$ $$ \beta = 90^o - 123.749^o $$ $$ \beta = -33.749^o $$

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

Rhombus Calculator