Understand and use the standard small angle approximations of sine, cosine and tangent: sin θ ≈ θ, cos θ ≈ 1 – θ²/2, tan θ ≈ θ where θ is in radians

Small angle approximations are a powerful tool in A-Level Mathematics, particularly in calculus and mechanics. When θ is measured in radians and is very small (typically |θ| < 0.1 radians), the trigonometric functions sin θ, cos θ, and tan θ can be approximated by simple polynomial expressions. These approximations simplify complex calculations, especially when dealing with limits, derivatives, and series expansions. For example, sin θ ≈ θ, cos θ ≈ 1 – θ²/2, and tan θ ≈ θ. Understanding these approximations is essential for solving problems involving pendulum motion, small oscillations, and geometric optics.

The approximations are derived from the Maclaurin series expansions of the trigonometric functions. For sin θ, the series is θ – θ³/6 + ...; for cos θ, it is 1 – θ²/2 + θ⁴/24 – ...; and for tan θ, it is θ + θ³/3 + ... . By truncating after the first term (or first two for cos), we obtain the standard small angle approximations. The key condition is that θ must be in radians, as the series expansions are based on radian measure. This topic is a prerequisite for understanding more advanced concepts like Taylor series and differential equations.

In the Edexcel A-Level specification, small angle approximations appear in the context of pure mathematics (e.g., evaluating limits like lim_{θ→0} sin θ/θ) and in mechanics (e.g., approximating sin θ ≈ θ for a pendulum with small amplitude). Mastery of this topic allows students to simplify expressions and solve problems that would otherwise be intractable. It also reinforces the importance of radian measure and the behaviour of functions near zero.