Question

Sin (x/2)= √2- sin (x/2) solve for the exact solutions in the interval (0, 2π)

Answer 1

`sin\left( (x)/(2) \right)=√(2)-sin\left( (x)/(2) \right)\implies sin\left( (x)/(2) \right)+sin\left( (x)/(2) \right)=√(2)\implies 2sin\left( (x)/(2) \right)=√(2) \\\\\\ sin\left( \cfrac{x}{2} \right)=\cfrac{√(2)}{2}\implies sin^(-1)\left[ sin\left( \cfrac{x}{2} \right) \right]=sin^(-1)\left( \cfrac{√(2)}{2} \right) \implies \cfrac{x}{2}= \begin{cases} (\pi )/(4)\\\\ (3\pi )/(4) \end{cases} \\\\[-0.35em] ~\dotfill`

`\cfrac{x}{2}=\cfrac{\pi }{4}\implies \boxed{x=\cfrac{\pi }{2}}~\hfill \cfrac{x}{2}=\cfrac{3\pi }{4}\implies \boxed{x=\cfrac{3\pi }{2}}`