Find the diagonal $ d_2 $ of a rhombus if side $a = 22$ and angle $\beta = 40^o$.

$d_2 = 49.2748$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ \beta = 40^o $ we have:

$$ \alpha + 40^o = 90^o $$ $$ \alpha = 90^o - 40^o $$ $$ \alpha = 50^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ a } $$

After substituting $ \alpha = 50^o $ and $ a = 22 $ we have:

$$ \sin( 50^o ) = \dfrac{ h }{ 22 } $$ $$ 0.766 = \dfrac{ h }{ 22 } $$$$ h = 0.766 \cdot 22 $$$$ h = 16.853 $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = 20^o $ and $ h = 16.853 $ we have:

$$ \sin \left( \frac{ 40^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 20^o ) = \dfrac{ 16.853 }{ d_2 } $$ $$ 0.342 = \dfrac{ 16.853 }{ d_2 } $$ $$ d_2 = \dfrac{ 16.853 }{ 0.342 } $$ $$ d_2 = 49.2748 $$

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