How to get $ \left(5+\sqrt{6}\right)$ from the expression
$$6\sqrt { 3 } \sin { \left( \frac { 2\pi }{ 3 } -\arcsin { \left( \frac { 5\sqrt { 3 } }{ 9 } \right) } \right) } $$
without using calculator for examination purpose?
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haqnatural
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Pallab
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by using identity $$\sin { \left( \alpha -\beta \right) =\sin { \alpha \cos { \beta -\sin { \beta \cos { \alpha } } } } } $$ we get $$6\sqrt { 3 } \sin { \left( \frac { 2\pi }{ 3 } -\arcsin { \left( \frac { 5\sqrt { 3 } }{ 9 } \right) } \right) } =6\sqrt { 3 } \left[ \sin { \frac { 2\pi }{ 3 } \cos { \left( \arcsin { \left( \frac { 5\sqrt { 3 } }{ 9 } \right) } \right) -\cos { \frac { 2\pi }{ 3 } \sin { \left( \arcsin { \left( \frac { 5\sqrt { 3 } }{ 9 } \right) } \right) } } } } \right] =\\ =6\sqrt { 3 } \left[ \frac { \sqrt { 3 } }{ 2 } \sqrt { 1-{ \left( \frac { 5\sqrt { 3 } }{ 9 } \right) }^{ 2 } } +\frac { 1 }{ 2 } \left( \frac { 5\sqrt { 3 } }{ 9 } \right) \right] =6\sqrt { 3 } \left[ \frac { \sqrt { 2 } }{ 6 } +\frac { 5\sqrt { 3 } }{ 18 } \right] =\color{blue}{\sqrt { 6 } +5}$$
haqnatural
- 21,578