Small Angle Approximation — Definition, Formula & Examples

Small angle approximation is the practice of replacing trigonometric functions with simpler expressions when the angle is close to zero: sin θ ≈ θ, cos θ ≈ 1, and tan θ ≈ θ, where θ is measured in radians.

For an angle θ expressed in radians with |θ| ≪ 1, the first-order Taylor expansions of sine and tangent about θ = 0 yield sin θ ≈ θ and tan θ ≈ θ, while the second-order expansion of cosine gives cos θ ≈ 1 − θ²/2, which is further approximated as cos θ ≈ 1 when only first-order accuracy is needed.

Key Formula

\sin\theta \approx \theta, \qquad \cos\theta \approx 1 - \frac{\theta^2}{2}, \qquad \tan\theta \approx \theta

Where:

$\theta$ = Angle measured in radians, where |θ| is much less than 1

How It Works

These approximations come from truncating the Maclaurin (Taylor) series of each trig function. For sine: sin θ = θ − θ³/6 + ⋯, so dropping the cubic and higher terms leaves sin θ ≈ θ. The approximation is remarkably accurate for small angles — at θ = 0.1 rad (about 5.7°), sin θ differs from θ by only about 0.17%. You must use radians; the approximation fails completely if θ is in degrees. A common rule of thumb is that the approximation stays within about 1% error for angles up to roughly 0.24 rad (about 14°).

Worked Example

Problem: A simple pendulum of length 2 m swings with a maximum angle of 0.05 radians. Use the small angle approximation to estimate sin(0.05) and find the horizontal displacement at the maximum angle.

Apply the approximation: Since 0.05 rad is very small, replace sin θ with θ.

\sin(0.05) \approx 0.05

Find horizontal displacement: The horizontal displacement of a pendulum is x = L sin θ. Substituting the approximation:

x \approx L\,\theta = 2 \times 0.05 = 0.10 \text{ m}

Check accuracy: The exact value is sin(0.05) = 0.049979..., so the approximation introduces an error of only about 0.04%.

\text{Error} = \frac{|0.05 - 0.049979|}{0.049979} \times 100\% \approx 0.04\%

Answer: sin(0.05) ≈ 0.05, giving a horizontal displacement of approximately 0.10 m.

Why It Matters

In physics, the small angle approximation turns the nonlinear pendulum equation into a solvable linear differential equation, producing the classic period formula T = 2π√(L/g). It appears throughout optics (Snell's law for paraxial rays), structural engineering (beam deflection), and astronomy (parallax calculations). In calculus courses, it provides a concrete example of linearization and Taylor series truncation.

Common Mistakes

Mistake: Using the approximation with angles measured in degrees instead of radians.

Correction: The formulas sin θ ≈ θ and tan θ ≈ θ are only valid when θ is in radians. An input of 5 degrees must first be converted to 5π/180 ≈ 0.0873 rad before applying the approximation.