Find the angle $ \alpha $ of a rhombus if side $a = 3 \sqrt{ 5 }$ and diagonal $d_2 = 3$.

$\alpha = 26.5254^o$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_2 = 3 $ and $ a = 3 \sqrt{ 5 } $ we have:

$$ d_1 ^ {\,2} + 3^2 = 4 \cdot \left(3 \sqrt{ 5 }\right)^2 $$ $$ d_1 ^ {\,2} + 9 = 180 $$ $$ d_1 ^ {\,2} = 180 - 9 $$ $$ d_1 ^ {\,2} = 171 $$ $$ d_1 = \sqrt{ 171 } $$$$ d_1 = 3 \sqrt{ 19 } $$

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 = 3 $ and $ d_1 = 3 \sqrt{ 19 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 3 }{ 3 \sqrt{ 19 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{\sqrt{ 19 }}{ 19 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{\sqrt{ 19 }}{ 19 } \right) $$ $$ \frac{ \alpha }{ 2 } = 13.2627^o $$$$ \alpha = 13.2627^o \cdot 2 $$$$ \alpha = 26.5254^o $$

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