Half-angle identities · trigonometric identity
sin(x/2) = ±√((1 − cos x)/2) — Half-angle identity for sine
When to use it
A half angle appears, or an integral contains √(1 − cos x). Always decide the sign from the quadrant of x/2 BEFORE writing the root — that decision is the whole problem.
Why it is true
Start from cos 2u = 1 − 2sin²u, solve for sin u and rename u = x/2. The square root is unavoidable, which is exactly why a sign question comes with it.
The full line-by-line version is on the proof page for half-angle identity for sine; the “how would I find this myself” version is in the derivation.
Where it 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.
- 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.
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
- Simplify √(1 − cos x) so it can be integrated√2·|sin(x/2)| — on 0 < x < 2π the absolute value is sin(x/2)
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 |
|---|---|---|---|
| 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 |
| 45° | 0.382683 | 0.382683 | yes |
| 53° | 0.446198 | 0.446198 | yes |
| 60° | 0.5 | 0.5 | yes |
| 67.5° | 0.55557 | 0.55557 | yes |
| 75° | 0.608761 | 0.608761 | yes |
| 90° | 0.707107 | 0.707107 | yes |
| 120° | 0.866025 | 0.866025 | yes |
| 135° | 0.92388 | 0.92388 | yes |
| 150° | 0.965926 | 0.965926 | yes |
| 180° | 1 | 1 | yes |
| 210° | 0.965926 | 0.965926 | yes |
| 240° | 0.866025 | 0.866025 | yes |
| 270° | 0.707107 | 0.707107 | yes |
| 300° | 0.5 | 0.5 | yes |
| 330° | 0.258819 | 0.258819 | yes |
Condition on the angle
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.
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
Writing ± and moving on. In an exercise you are expected to choose: if x/2 is in the third quadrant, sin(x/2) is negative and the answer is the negative root.
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
- cos(x/2) = ±√((1 + cos x)/2)Half-angle identity for cosinedetailsproof
- tan(x/2) = (1 − cos x)/sin xHalf-angle identity for tangent (form 1)detailsproof
- cos 2x = 1 − 2 sin²xDouble-angle identity for cosine (form 3)detailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
Category hub: Half-angle identities · all identities: /identities