Half-angle identities · derivation ·proof

Where cos(x/2) = ±√((1 + cos x)/2) comes from

Same quadrant rule, cosine's signs. Cosine is positive in Q1 and Q4 — so x/2 must be there for the + root.

Building it step by step

1. Magnitude: √((1 + cos x)/2)
   From the identity.
2. Where is x/2?
   Q1/Q4 → +, Q2/Q3 → −.
3. −180° < x < 180° puts x/2 in Q1/Q4
   Which is why the + version is quoted with that interval.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from cos(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 cosine page.

sampled at 800 valid angles inside its stated intervalconditional: x between -180° and 180°Source: OpenStax Precalculus, Ch. 7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error