For angles like \(22.5^\circ,15^\circ,\) and \(75^\circ\), use half-angle identities:
\[
\sin\frac{\theta}{2}
=
\sqrt{\frac{1-\cos\theta}{2}}
\]
and
\[
\cos\frac{\theta}{2}
=
\sqrt{\frac{1+\cos\theta}{2}}.
\]
Step 1: Identify the half-angle formula for sine. We know that $ \sin\frac{\theta}{2} = \sqrt{\frac{1 - \cos\theta}{2}} $ (taking the positive root since the angle $ 22.5^\circ $ is in the first quadrant where sine is positive). Step 2: Recognise that 22.5 degrees is half of 45 degrees. $ 22\frac{1}{2}^\circ = \frac{45^\circ}{2} $. So we use the half-angle formula with $ \theta = 45^\circ $. Step 3: Substitute into the formula. \[ \sin 22\tfrac{1}{2}^\circ = \sqrt{\frac{1 - \cos 45^\circ}{2}} \] Step 4: Substitute the value of cos 45 degrees. $ \cos 45^\circ = \frac{\sqrt{2}}{2} $. So: \[ \sin 22\tfrac{1}{2}^\circ = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} \] Step 5: Confirm this is a valid simplification. We combined the fractions step by step: $ 1 - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} $, then dividing by 2 gives $ \frac{2 - \sqrt{2}}{4} $. The square root of that is the answer. Step 6: State the final answer. \[ \boxed{\sqrt{\frac{2 - \sqrt{2}}{4}}} \]