The final line of my attempt contains the 2 answers listed in the book, but I'm not sure what the basis for excluding all the other values under $\displaystyle 180^{\circ}$ is? I have a feeling my approach is slightly incorrect here. Can anyone help me clear this up?

Many thanks.

Solve the equation $\displaystyle \sin2x=\frac{1}{\sqrt{2}}$ for $\displaystyle 0\leq x\leq\pi$.

Q.

Attempt:$\displaystyle \sin2x=\frac{1}{\sqrt{2}}\rightarrow2x=45^{\circ}$. Sin is positive in the 1st & 2nd quadrants.

$\displaystyle \sin2x=\frac{1}{\sqrt{2}}\rightarrow2x=0^{\circ}+4 5^{\circ}=45^{\circ}$ or $\displaystyle 180^{\circ}-45^{\circ}=135^{\circ}$

Since the angle is $\displaystyle 2x$ add $\displaystyle 90^{\circ}$ & $\displaystyle 270^{\circ}$ to each angle (i.e. $\displaystyle 2(45^{\circ})$ & $\displaystyle 2(135^{\circ}$):

$\displaystyle 2x=45^{\circ},135^{\circ},315^{\circ}$ or $\displaystyle 2x=135^{\circ},225^{\circ},405^{\circ}$

$\displaystyle x=22.5^{\circ},67.5^{\circ},157.5^{\circ}$ or $\displaystyle x=67.5^{\circ},112.5^{\circ},202.5^{\circ}$

(From text book): $\displaystyle 22.5^{\circ},67.5^{\circ}$

Ans.