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Pure · Trigonometry

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Small angle approximations

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

Pure · Trigonometry

Small angle approximations

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

Why it works

When an angle θ\theta is small and measured in radians, the trig functions flatten out into simple polynomials: sin⁡θ≈θ,tan⁡θ≈θ,cos⁡θ≈1−12θ2.\sin\theta \approx \theta, \qquad \tan\theta \approx \theta, \qquad \cos\theta \approx 1 - \tfrac12\theta^2.

Where they come from. Picture a tiny sector of a unit circle. The arc is θ\theta, and for a small angle the arc, the chord, and the vertical height sin⁡θ\sin\theta are almost indistinguishable — so sin⁡θ≈θ\sin\theta\approx\theta. Likewise tan⁡θ≈θ\tan\theta\approx\theta. Cosine is the horizontal side, which barely shrinks from 11; the leading correction is quadratic, giving cos⁡θ≈1−12θ2\cos\theta\approx 1-\tfrac12\theta^2. (These are exactly the first terms of the series sin⁡θ=θ−θ36+⋯\sin\theta=\theta-\tfrac{\theta^3}{6}+\cdots and cos⁡θ=1−θ22+⋯\cos\theta=1-\tfrac{\theta^2}{2}+\cdots.)

Two things to watch. They hold only in radians (in degrees sin⁡θ\sin\theta is nowhere near θ\theta), and only for small θ\theta — the bigger θ\theta is, the worse they get.

Keep the multiplier. For a multiple of θ\theta, scale inside the function: sin⁡kθ≈kθ,tan⁡kθ≈kθ,cos⁡kθ≈1−12(kθ)2=1−12k2θ2.\sin k\theta \approx k\theta, \qquad \tan k\theta \approx k\theta, \qquad \cos k\theta \approx 1 - \tfrac12(k\theta)^2 = 1 - \tfrac12 k^2\theta^2.

How they're used. Two standard exam tasks: (1) "show that f(θ)≈a+bθ+cθ2f(\theta)\approx a+b\theta+c\theta^2" — replace each trig function by its approximation and collect terms; (2) "show that f(θ)g(θ)→\dfrac{f(\theta)}{g(\theta)}\to constant for small θ\theta" — replace top and bottom by their leading terms and cancel.