Find the area $ A $ of a rhombus if diagonal $d_1 = 5$ and angle $\beta = 60^o$.

$A = 3.2352$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ \beta = 60^o $ we have:

$$ \alpha + 60^o = 90^o $$ $$ \alpha = 90^o - 60^o $$ $$ \alpha = 30^o $$

STEP 2: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ \alpha = 15^o $ and $ d_1 = 5 $ we have:

$$ \sin \left( \frac{ 30^o }{ 2 } \right) = \dfrac{ d_2 }{ 5 } $$ $$ \sin( 15^o ) = \dfrac{ d_2 }{ 5 } $$ $$ 0.2588 = \dfrac{ d_2 }{ 5 } $$$$ d_2 = 0.2588 \cdot 5 $$$$ d_2 = 1.2941 $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $ d_1 = 5 $ and $ d_2 = 1.2941 $ we have:

$$ A = \dfrac{ 5 \cdot 1.2941 }{ 2 }$$$$ A = \dfrac{ 6.4705 }{ 2 } $$$$ A = 3.2352 $$

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