Half-angle identities · trigonometric identity
cos(x/2) = ±√((1 + cos x)/2) — Half-angle identity for cosine
When to use it
Half-angle cosine, or √(1 + cos x) under an integral. Same sign discipline as the sine version: the quadrant of x/2 decides.
Why it is true
Start from cos 2u = 2cos²u − 1, solve it for cos²u, take the root and rename u as x/2. The root is where the sign question comes from — the square loses the information about which half-plane x/2 lives in.
The full line-by-line version is on the proof page for half-angle identity for cosine; the “how would I find this myself” version is in the derivation.
Where it comes from
Same quadrant rule, cosine's signs. Cosine is positive in Q1 and Q4 — so x/2 must be there for the + root.
- Magnitude: √((1 + cos x)/2)From the identity.
- Where is x/2?Q1/Q4 → +, Q2/Q3 → −.
- −180° < x < 180° puts x/2 in Q1/Q4Which is why the + version is quoted with that interval.
Worked examples
- Given cos x = −3/5 with 180° < x < 270°, find sin(x/2) and cos(x/2)sin(x/2) = √(4/5) = 2/√5 and cos(x/2) = −1/√5
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 1 | 1 | yes |
| 7.5° | 0.997859 | 0.997859 | yes |
| 15° | 0.991445 | 0.991445 | yes |
| 18° | 0.987688 | 0.987688 | yes |
| 22.5° | 0.980785 | 0.980785 | yes |
| 30° | 0.965926 | 0.965926 | yes |
| 37° | 0.948324 | 0.948324 | yes |
| 45° | 0.92388 | 0.92388 | yes |
| 53° | 0.894934 | 0.894934 | yes |
| 60° | 0.866025 | 0.866025 | yes |
| 67.5° | 0.83147 | 0.83147 | yes |
| 75° | 0.793353 | 0.793353 | yes |
| 90° | 0.707107 | 0.707107 | yes |
| 120° | 0.5 | 0.5 | yes |
| 135° | 0.382683 | 0.382683 | yes |
| 150° | 0.258819 | 0.258819 | yes |
Condition on the angle
x between -180° and 180° — the + root holds for −180° < x < 180° (x/2 in the right half-plane). At x = 270°, for instance, cos 135° = −√2/2 while the + root gives +√2/2 — the identity fails, which our sampler verifies.
Outside that interval this page shows the counterexample: the formula as written stops being true, which is why textbooks put a ± in front of it.
The mistake students make
Confusing which root goes with which sign inside the radical: 1 − cos x belongs to sine, 1 + cos x belongs to cosine. Memory hook: cosine is the 'plus' one.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for half-angle identities.
Related identities
- sin(x/2) = ±√((1 − cos x)/2)Half-angle identity for sinedetailsproof
- tan(x/2) = (1 − cos x)/sin xHalf-angle identity for tangent (form 1)detailsproof
- cos 2x = 2 cos²x − 1Double-angle identity for cosine (form 2)detailsproof
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
Category hub: Half-angle identities · all identities: /identities