Library / Foundations: Maths / FMA-23
Foundations: Maths
Small-angle approximation
How accurate sin x ≈ x is for a given angle.
- Ref
- FMA-23
- Rev
- v1
- Field
- Foundations: Maths
- Result
- ε [%]
Accuracy improves as x approaches zero; x must be in radians. The calculator shows the percentage error made by replacing sin x with x.
For small angles measured in radians, the sine is almost equal to the angle itself, which makes many problems linear.
Calculator
- Error in tan x ≈ x−1.017 %
- Error in cos x ≈ 1 − x²/2−0.003922 %
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.
- Below about 10° the approximation is within about 0.5%; at 30° it is already about 5%.
- The same idea linearises the pendulum and small deflections.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Angle | 1 to 30 ° | 0.005077 | 4.72 | 4.715 | +10% | +0.107 pp |
Sweep of Angle
| Angle (°) | Error in sin x ≈ x (%) |
|---|---|
| 1 | 0.005077 |
| 1.53 | 0.01188 |
| 2.34 | 0.02781 |
| 3.58 | 0.06511 |
| 5.477 | 0.1525 |
| 8.379 | 0.3573 |
| 12.82 | 0.8391 |
| 19.61 | 1.979 |
| 30 | 4.72 |
Sweep one variable
Variables and how each one contributes
Error in sin x ≈ x
What the equation gives you. Everything else feeds it.
Angle
The angle, entered in degrees and converted to radians. The error grows with x², so doubling the angle quadruples it.
Worked example with the values above
- Convert to radians: 10° = 0.174533 rad.
- sin x = 0.173648.
- Error = (x − sin x)/sin x = 0.51 %.
Assumptions and limits
- Angles in radians.
- Angle small compared with 1 radian.
Common mistakes
- Using degrees without converting.
- Applying it at large angles.
Statistics and uncertainty
Sensor and sight alignment angles are often a few degrees, where the small-angle error is far below the measurement uncertainty.
Where it is used
- Pendulum and vibration models.
- Small pointing-error and misalignment estimates.
Maths you need
- Radians: convert from degrees
- Sine and tangent: for angles