What is the small-angle approximation for sin θ?

sin θ ≈ θ, where θ is in radians. The error is approximately θ³/6 for small angles.

Why must θ be in radians?

The derivative of sin θ is cos θ only when θ is in radians. In degrees, the derivative includes a factor of π/180, and the approximation sin θ ≈ θ gives nonsense results.

How accurate is the approximation at 10°?

10° is about 0.175 rad. The error in sin θ is roughly 0.0009 (0.09% relative error), and the error in the pendulum period is about 0.2%.

What is the error bound for the linearization?

Taylor’s inequality for n = 1 gives |E(θ)| ≤ (M/2)θ², where M is the maximum of |f''| on the interval.

When does the approximation fail completely?

At angles above about 0.5 rad (29°), the error for sin θ exceeds 4% and grows rapidly.

Small-Angle Approximation: sin θ ≈ θ, tan θ ≈ θ

The Small-Angle Approximation

A pendulum bob swings. You measure the period, plug numbers into T = 2π√(L/g), and get a prediction that matches your stopwatch within a few milliseconds. That formula works because of the small-angle approximation: for angles under about 0.2 rad (roughly 12°), sin θ ≈ θ, and the messy nonlinear pendulum equation becomes a simple harmonic oscillator. The approximation is a linearization of sin θ at θ = 0, the tangent line of the sine curve at the origin. Use it when you need a quick, analytic result and can live with a small, predictable error.

The Approximations as Linearizations at 0

The small-angle approximation is a direct application of the linearization formula L(x) = f(a) + f’(a)(x, a) with a = 0. For f(θ) = sin θ, f’(0) = cos 0 = 1, so L(θ) = 0 + 1·θ = θ. For tan θ, f’(0) = sec² 0 = 1, giving the same linearization. For cos θ, f’(0) =, sin 0 = 0, so the first-order approximation would be cos θ ≈ 1; the more useful second-order approximation cos θ ≈ 1, θ²/2 comes from the Taylor polynomial of degree 2. All three are tangent lines (or tangent parabolas for cos) at θ = 0. The error grows with the distance from the point of tangency, as Taylor’s inequality for n = 1 predicts: |E(θ)| ≤ (M/2)θ², where M bounds |f’’| on the interval. For sin θ, the next term in the series is, θ³/6, so the error is roughly θ³/6 for small angles.

Radians Only

The small-angle approximation is dimensionally incorrect in degrees. If you plug θ = 1° into sin θ ≈ θ, you get sin 1° ≈ 1, while the true value is about 0.0175. The linearization assumes the derivative of sin θ is cos θ, a formula valid only when θ is in radians. Convert any angle to radians before applying the approximation: multiply degrees by π/180. Physics textbooks (Halliday/Resnick, for instance) enforce this rule in their pendulum derivations, the equation d²θ/dt² =, (g/L)θ follows from sin θ ≈ θ only if θ is in radians.

Accuracy Table by Angle

Table: Small-Angle Accuracy for sin, cos, and tan

Small-Angle Accuracy for Sin Θ, Cos Θ, and Tan Θ

θ (rad)	sin θ exact	sin θ approx	sin error	cos θ exact	cos θ approx	cos error	tan θ exact	tan θ approx	tan error
0.01	0.009999833	0.01	2×10⁻⁷ over	0.999950	0.99995	0 exact to 6 dp	0.010000	0.01	3×10⁻⁷ over
0.05	0.049937	0.05	6.3×10⁻⁵ over	0.99875	0.99875	0 exact to 5 dp	0.05004	0.05	4×10⁻⁵ over
0.10	0.099334	0.10	6.6×10⁻⁴ over	0.9950	0.9950	0 exact to 4 dp	0.1003	0.10	3×10⁻⁴ over
0.20	0.196	0.20	4×10⁻³ over	0.980	0.980	0 exact to 3 dp	0.203	0.20	0.003 over
0.50	0.479	0.50	0.021 over	0.877	0.875	0.002 under	0.546	0.50	0.046 over
1.00	0.841	1.00	0.159 over	0.540	0.500	0.040 under	1.557	1.00	0.557 over

Simple Pendulum Example

The classic physics application is the simple pendulum. The restoring force depends on sin θ, where θ is the angular displacement from vertical. The exact equation of motion is d²θ/dt² =, (g/L) sin θ. For small angles, sin θ ≈ θ, so the equation becomes d²θ/dt² =, (g/L)θ, a linear differential equation with the well-known solution for the period T = 2π√(L/g). This derivation appears in first-year physics texts such as Halliday/Resnick’s Fundamentals of Physics.

How the Error Affects the Period

The exact period of a pendulum is given by an elliptic integral: T = 2π√(L/g) · (2/π)K(k), where k = sin(θ₀/2) and K(k) is the complete elliptic integral of the first kind. For an initial amplitude of 5°, the small-angle approximation underestimates the period by about 0.05%. At 10° the error is roughly 0.2%; at 20° it is about 0.8%; and at 45° the error reaches approximately 1.8%. Whether these errors matter depends on your application. In a classroom demonstration, the 0.2% error at 10° is negligible. For a precision timekeeping instrument, even 0.05% may be unacceptable, a pendulum clock gaining 0.05% runs about 43 seconds fast per day.

When the Approximation Breaks Down

The approximation fails when the angle is too large for the tangent line to remain close to the curve. At θ = 0.4% relative to the true value.557 (36% relative error). Two failure modes are common: using degrees instead of radians, and assuming the approximation holds for any angle labeled “small” without checking the error. The error bound from Taylor’s inequality gives a worst-case guarantee, but the actual error may be smaller, and the bound itself depends on the second derivative, which for sin θ is, sin θ, with a maximum magnitude of 1 on the interval [0, θ]. For θ = 0.2 rad, the bound is (1/2)(0.2)² = 0.

Who the Approximation Suits and Who Should Skip It

The small-angle approximation suits AP Calculus AB/BC students who need to compute linearizations by hand and interpret over- or underestimates using concavity; Calculus III students extending the idea to tangent-plane approximations for functions of two variables; and physics students using it in pendulum and optics derivations. Anyone needing exact function values, numerical root-finding (Newton’s method), or higher-order approximations (Taylor polynomials beyond n = 1) should use a Taylor series calculator or a root-finding resource instead.