Small Angle Approximations

When theta is small (measured in radians), the trigonometric functions can be approximated by simpler expressions. These approximations are given in the Edexcel formula booklet and are tested at A-Level (9MA0).

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The Three Approximations

For small theta (in radians):

sin(theta) ≈ theta

cos(theta) ≈ 1 - theta²/2

tan(theta) ≈ theta

These become more accurate as theta approaches 0.

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Understanding the Approximations

Function	Approximation	Why it works
sin(theta)	theta	The sine curve is almost linear near the origin
cos(theta)	1 - theta²/2	This is the start of the Taylor series for cosine
tan(theta)	theta	For small angles, tan ≈ sin ≈ theta

Numerical Evidence

theta (radians)	sin(theta)	theta	cos(theta)	1 - theta²/2	tan(theta)
0.01	0.009999833	0.01	0.99995	0.99995	0.01000003
0.1	0.09983	0.1	0.99500	0.99500	0.10033
0.2	0.19867	0.2	0.98007	0.98000	0.20271
0.5	0.47943	0.5	0.87758	0.87500	0.54630