<s>[INST] Solving Trigonometric Equation 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?
 [/INST] by using  identity $$\sin { \left( \alpha -\beta  \right) =\sin { \alpha \cos { \beta -\sin { \beta \cos { \alpha  }  }  }  }  } $$ we get $$6\sqrt { 3 } \sin { \left( \frac { 2\pi  }{ 3 } -\arcsin { \left( \frac { 5\sqrt { 3 }  }{ 9 }  \right)  }  \right)  } =6\sqrt { 3 } \left[ \sin { \frac { 2\pi  }{ 3 } \cos { \left( \arcsin { \left( \frac { 5\sqrt { 3 }  }{ 9 }  \right)  }  \right) -\cos { \frac { 2\pi  }{ 3 } \sin { \left( \arcsin { \left( \frac { 5\sqrt { 3 }  }{ 9 }  \right)  }  \right)  }  }  }  }  \right] =\\ =6\sqrt { 3 } \left[ \frac { \sqrt { 3 }  }{ 2 } \sqrt { 1-{ \left( \frac { 5\sqrt { 3 }  }{ 9 }  \right)  }^{ 2 } } +\frac { 1 }{ 2 } \left( \frac { 5\sqrt { 3 }  }{ 9 }  \right)  \right] =6\sqrt { 3 } \left[ \frac { \sqrt { 2 }  }{ 6 } +\frac { 5\sqrt { 3 }  }{ 18 }  \right] =\color{blue}{\sqrt { 6 } +5}$$
</s>