Small Angle Approximations

This lesson covers the small angle approximations for sin θ, cos θ, and tan θ when θ is small and measured in radians. These approximations simplify expressions and are used in limit calculations, mechanics, and numerical analysis. They are part of the AQA A-Level Mathematics specification 7357.

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The Three Approximations

When θ is small and measured in radians:

sin θ ≈ θ
cos θ ≈ 1 − θ²/2
tan θ ≈ θ

These approximations become more accurate as θ gets closer to zero, and they are exact in the limit as θ → 0.

Crucially, θ must be in radians. The approximations do not work with degrees.

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Why Do These Work?

The sin θ ≈ θ Approximation

Consider the unit circle. For a small angle θ (in radians), the arc length subtended is θ (since s = rθ = 1 × θ), and the opposite side of the triangle (which equals sin θ) is approximately equal to the arc length for small angles. As the angle shrinks, the chord and the arc become indistinguishable.

More formally, from the Maclaurin series (Taylor series at 0):