A Level Maths: Trigonometry: Small Angle Approximations and Inverse Trig Functions Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Trigonometry — foundational trig
- Trigonometry — Radians, Circle Sectors and Triangles}}: radian measure is required for the approximations
- Functions — inverse functions and restricted domains
A Level Maths trigonometry with small angle approximations and inverse trig functions rounds out the Year 2 trigonometry block. Small angle approximations give you simple polynomial approximations for trig functions near zero, useful for limits and modelling. Inverse trig functions ($\arcsin$, $\arccos$, $\arctan$) are the inverses of the standard trig functions on restricted domains.
For small angle approximations (in radians), you use $\sin(x) \approx x$, $\cos(x) \approx 1 – \tfrac{x^2}{2}$, and $\tan(x) \approx x$ when $x$ is close to zero. These let you approximate complicated expressions for small inputs — for example, $\frac{1 – \cos(3x)}{\sin(4x)}$ for small $x$ simplifies via these approximations. You also meet the inverse trig functions: $\arcsin$ (or $\sin^{-1}$) with domain $[-1, 1]$ and range $[-\pi/2, \pi/2]$; $\arccos$ with domain $[-1, 1]$ and range $[0, \pi]$; and $\arctan$ with domain all real numbers and range $(-\pi/2, \pi/2)$. You sketch their graphs (reflections of $\sin$, $\cos$, $\tan$ in $y = x$, with appropriate domain restrictions) and use them to express exact angles.
Trigonometry: small angle approximations and inverse trig functions is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: small angle approximations only work in RADIANS — using them with degrees is wrong; the inverse trig functions are NOT $\frac{1}{\sin}$, $\frac{1}{\cos}$, $\frac{1}{\tan}$ (those are $\sec$, $\csc$, $\cot$) — $\sin^{-1}$ is the inverse function, not the reciprocal; the range of $\arcsin$ is $[-\pi/2, \pi/2]$, not $(0, \pi)$ — getting the principal value range wrong is common; and small angle approximations are only valid NEAR ZERO — they do not work for arbitrary angles.