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Taylor series and small angles
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Approximate functions near a point, e.g. sinθ ≈ θ.
sinθ ≈ θ − θ³/6A smooth function near a point can be approximated by a polynomial: sinθ ≈ θ − θ³/6, cosθ ≈ 1 − θ²/2, (1+x)ⁿ ≈ 1 + nx. For small angles in radians, sinθ ≈ tanθ ≈ θ, which turns hard nonlinear equations into solvable ones, such as the small-angle pendulum.
At θ = 0.1 rad, sinθ = 0.09983, only 0.17% less than θ.
Angles must be in radians, and the approximation fails for larger angles; at 30° the error is about 4.5%.
In more detail
A Taylor series writes a function as a sum of powers of x, with coefficients given by its derivatives. For small angles, sin θ is approximately θ, and cos θ is approximately 1 minus θ squared over two.
Real-world uses
- The small-angle approximation makes a simple pendulum behave like a simple harmonic oscillator.
- Optics uses paraxial approximations, based on series expansions, to model lenses and mirrors.
- A pendulum with a 5-degree swing period is almost identical to the small-angle formula, which makes the design of clocks much simpler.
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Frequently asked questions
Why does the small-angle approximation work?
For small angles, the first term of the series dominates, so higher-order corrections are tiny.
How small must the angle be?
For θ in radians, 0.1 rad is about 6 degrees and the approximation is accurate to roughly half a percent.
Is the small-angle approximation always valid?
No. It works well for angles below about 10 degrees, but the error grows quickly for larger swings, so check the size of the angle first.
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Partial derivatives · Gradient, divergence, curl · Differential equations in physics · Complex numbers and phasors · Graphs and slopes · Curve fitting and linearisation