Half-angle identities · proof ·what it is and when to use it
Proof: sin(x/2) = ±√((1 − cos x)/2)
Half-angle identity for sine — proved by solve cos 2u for sin u. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Rearranging the double-angle cosine, then a change of name.
- cos 2u = 1 − 2sin²uDouble-angle form 3. · Double-angle identity for cosine (form 3)
- sin²u = (1 − cos 2u)/2Solve for sin²u.
- sin u = ±√((1 − cos 2u)/2)Take the root; sign unknown until the quadrant of u is known.
- u = x/2 → sin(x/2) = ±√((1 − cos x)/2)Rename 2u as x.
With 0° < x < 360°, the half angle x/2 lies between 0° and 180° where sine is positive — so the + root is correct on that interval and only there.
Where the proof stops applying
Requires x between 0° and 360° — the + root is the valid one for 0° < x < 360° (x/2 in the upper half-plane); outside that interval sin(x/2) is negative and the identity needs the − root.
This is not a technicality: outside the interval, the statement as written is false, and the sign has to be chosen by hand.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 0.065403 | 0.065403 | yes |
| 15° | 0.130526 | 0.130526 | yes |
| 18° | 0.156434 | 0.156434 | yes |
| 22.5° | 0.19509 | 0.19509 | yes |
| 30° | 0.258819 | 0.258819 | yes |
| 37° | 0.317305 | 0.317305 | yes |
Related
- How would I find sin(x/2) = ±√((1 − cos x)/2) myself? — the derivation, which is a different question from the proof.
- Half-angle identity for sine: when to use it — the practical side.
- All half-angle identities · proof index