Library / Geometry, coordinates, surveying and trigonometry / GEO-44
Geometry, coordinates, surveying and trigonometry
Positive half-angle sine
Sine of half an angle from the cosine of the angle, c = cos θ, for θ between 0 and 2π so that the half-angle sine is not negative.
- Ref
- GEO-44
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- \sin(θ/2)
The half-angle formula sin(θ/2) = √[(1 − cos θ)/2] follows from cos θ = 1 − 2 sin²(θ/2). The positive root is right for θ from 0 to 2π, where θ/2 lies between 0 and π and the sine is not negative. Outside that range the quadrant of θ/2 decides the sign.
Sine of half an angle from the cosine of the angle, c = cos θ, for θ between 0 and 2π so that the half-angle sine is not negative. Outside that range the quadrant sets the sign.
Calculator
- Half-angle cosine, √((1 + c)/2)0.5
- Half-angle in degrees (θ in [0°, 360°])60 °
- Angle θ in degrees120 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- If cos θ = −0.5, then θ = 120° (in the first half-turn) and sin(θ/2) = 0.866.
- At cos θ = −0.5, θ = 120°, and half of it is 60°, whose sine is √3/2 ≈ 0.866.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| c Cosine of the angle, cos θ | −1 to 1 | 0 | 1 | 1 | +0.04 | −1.34 % |
Sweep of Cosine of the angle, cos θ
| Cosine of the angle, cos θ | Sine of the half-angle |
|---|---|
| −1 | 1 |
| −0.75 | 0.9354 |
| −0.5 | 0.866 |
| −0.25 | 0.7906 |
| 0 | 0.7071 |
| 0.25 | 0.6124 |
| 0.5 | 0.5 |
| 0.75 | 0.3536 |
| 1 | 0 |
Sweep one variable
Variables and how each one contributes
Sine of the half-angle
What the equation gives you. Everything else feeds it.
Cosine of the angle, cos θ
The cosine of the angle θ, a plain number from −1 to 1. Raises sin(θ/2) as c falls, because a larger angle has a larger half-angle sine. Raising it by 0.05 (to −0.45) moves the result by −1.68 %, from 0.866 to 0.8515.
Worked example with the values above
- Subtract from 1: 1 − c = 1 − (−0.5) = 1.5.
- Halve it: 0.75.
- sin(θ/2) = √0.75 = 0.86603.
Assumptions and limits
- The inputs are the exact sine, cosine or tangent values of a real angle.
- Where a square root is taken, the stated range of the angle fixes the sign.
Common mistakes
- Using the formula when the tangent is undefined, such as at 90°.
- Taking the positive square root outside the range where the sign is positive.
- Mixing degrees and radians when using a calculator for the final angle.
Statistics and uncertainty
Identities are exact, so the only error is numerical. Tangent values near a pole (uv close to 1) magnify rounding, and a square root of a number near zero loses digits.
Where it is used
- Simplifying expressions and solving trigonometric equations in mathematics and physics.
- Combining two sinusoids of the same frequency into one wave, as in phasors, AC circuits and vibration.
Maths you need
- Sine, cosine and tangent: ratios from a right triangle or the unit circle
- Pythagorean identity: sin² + cos² = 1
- Phase: the horizontal shift of a wave