Half-angle identities · derivation ·proof
Where sin(x/2) = ±√((1 − cos x)/2) comes from
Quadrant decides the sign. The mechanical part is easy; the sign is the judgement call, and it is made from the quadrant of the HALF angle, not of x.
sin(x/2) = ±√((1 − cos x)/2)
Building it step by step
- Ignore the sign, write √((1 − cos x)/2)Magnitude from the identity.
- Ask: where is x/2?The half angle is the one whose sine you are computing.
- x/2 in Q1 or Q2 → +; in Q3 or Q4 → −Sine sign by quadrant.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(x/2) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the half-angle identity for sine page.