Find the diagonal $ d_2 $ of a rhombus if side $a = 338$ and angle $\alpha = \frac{ 73 }{ 2 }^o$.

$d_2 = 446.6803$

STEP 1: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = \frac{ 73 }{ 2 }^o $ we have:

$$ \frac{ 73 }{ 2 }^o + \beta = 90^o $$ $$ \beta = 90^o - \frac{ 73 }{ 2 }^o $$ $$ \beta = \frac{ 107 }{ 2 }^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = \frac{ 73 }{ 2 }^o $ and $ a = 338 $ we have:

$$ \sin( \frac{ 73 }{ 2 }^o ) = \dfrac{ h }{ 338 } $$ $$ 0.5948 = \dfrac{ h }{ 338 } $$$$ h = 0.5948 \cdot 338 $$$$ h = 201.0501 $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ \beta = \frac{ 107 }{ 4 }^o $ and $ h = 201.0501 $ we have:

$$ \sin \left( \frac{ \frac{ 107 }{ 2 }^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( \frac{ 107 }{ 4 }^o ) = \dfrac{ 201.0501 }{ d_2 } $$ $$ 0.4501 = \dfrac{ 201.0501 }{ d_2 } $$ $$ d_2 = \dfrac{ 201.0501 }{ 0.4501 } $$ $$ d_2 = 446.6803 $$

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