Find the diagonal $ d_2 $ of a rhombus if side $a = 4$ and angle $\alpha = 34^o$.

$d_2 = 4.7644$

STEP 1: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 34^o $ we have:

$$ 34^o + \beta = 90^o $$ $$ \beta = 90^o - 34^o $$ $$ \beta = 56^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 34^o $ and $ a = 4 $ we have:

$$ \sin( 34^o ) = \dfrac{ h }{ 4 } $$ $$ 0.5592 = \dfrac{ h }{ 4 } $$$$ h = 0.5592 \cdot 4 $$$$ h = 2.2368 $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ \beta = 28^o $ and $ h = 2.2368 $ we have:

$$ \sin \left( \frac{ 56^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 28^o ) = \dfrac{ 2.2368 }{ d_2 } $$ $$ 0.4695 = \dfrac{ 2.2368 }{ d_2 } $$ $$ d_2 = \dfrac{ 2.2368 }{ 0.4695 } $$ $$ d_2 = 4.7644 $$

Rhombus Calculator