Small Angle Approximation Calculator

This small angle approximation calculator shows how the trigonometric functions simplify when the angle is small and measured in radians. For a small angle x, sine is approximately x, tangent is approximately x, and cosine is approximately one minus x squared over two. These approximations are the basis of pendulum physics, optics, and many engineering derivations, because they turn awkward trig into simple algebra with very little error for small angles. Enter an angle in radians and the calculator shows the approximation for sine, the exact value, and the size of the error so you can see how good the approximation is. As a rule of thumb the error in sin x is well under one percent for angles below about 0.24 radians (14 degrees). The formula, a worked example and the assumptions are below.

Angle in radians

Exact sin

Approx error

The formula

For a small angle x in radians: sin(x) approx x, tan(x) approx x, and cos(x) approx 1 - x^2 / 2. The approximations improve as x approaches zero, and the angle must be in radians, not degrees.

Worked example

At x = 0.1 radians, the approximation gives sin x approx 0.1, while the exact sine is about 0.099833, an error of about 0.000167. Enter 0.1 above to confirm.

Frequently asked questions

When is the small angle approximation valid?

For small angles in radians. The smaller the angle, the better. Below about 0.24 radians (14 degrees) the error in sin x approx x is under one percent.

Must the angle be in radians?

Yes. The approximation sin x approx x only holds when x is in radians, not degrees.

What is the cosine approximation?

cos x approx 1 - x squared over two for small x in radians.

Who this calculator is for

This calculator is for physics and engineering students working with pendulums, optics or linearised models.

What this calculator assumes

You enter the angle in radians.
The approximations apply for small angles near zero.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions at once.
Degrees to Radians Calculator: convert angle units.
Right Angle Triangle Calculator: solve a right triangle.

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