What are the small-angle approximations?
For small angles in radians, sin x ≈ x, tan x ≈ x and cos x ≈ 1 − x²/2. The smaller the angle, the better the fit.
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Small-Angle Approximation
Compare sin x, tan x and cos x with their small-angle approximations (x, x and 1 − x²/2) and see the error, for small angles in radians.
sin x ≈ x, tan x ≈ x, cos x ≈ 1 − x²/2 (x in radians)
Frequently asked questions
Why must the angle be in radians?
The approximations come from the functions' series expansions in radians. In degrees they are simply wrong, so this calculator works in radians.
Can you show a worked example?
At x = 0.1 rad, sin x = 0.09983… while the approximation gives 0.1 — an error of about 0.17%. At x = 0.01 the error is far smaller still.
How small is 'small'?
There is no hard cut-off, but below about 0.1–0.2 rad (roughly 6°–12°) the error is well under 1% for sine and tangent.
Where are they used?
The simple pendulum (sin θ ≈ θ), optics (small-angle diffraction and lens formulae) and linearising equations in physics and engineering.