IA idea · Pure maths, number & proof
How does a calculator work out sin(x)?
Research question
How many terms of the Maclaurin series are needed to compute sin x to 10 decimal places for any x, and how do angle-reduction identities make the series efficient?
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Why it makes a good exploration
Calculators do not store tables: they compute. Exploring Maclaurin series, error bounds and the trick of reducing any angle to a small one shows how pure mathematics becomes an algorithm.
The mathematics you'll need
- Maclaurin series of sin x (HL; explain for SL)
- Alternating series error bounds
- Trigonometric identities for angle reduction
- Counting operations to compare methods
Where the data comes from
No data needed; compare your results with a calculator.
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Derive the Maclaurin series.
- Measure error for different numbers of terms and angles.
- Use identities to reduce angles to [0, π/4].
- Count terms needed with and without reduction.
- Reflect on CORDIC and other methods real devices use.
Pitfalls that cost marks
- Error estimates without the bound.
- Degrees vs radians.
- Only numerical experiments.
Showing personal engagement
- Write your own sin function and compare with the calculator.
- Find the worst-case angle for your method.
- Explain CORDIC in outline.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Maclaurin series of sin x (HL; explain for SL); Alternating series error bounds |
| AA HL | Good fit | Maclaurin series of sin x (HL; explain for SL); Alternating series error bounds |
| AI SL | Not a natural fit | The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach. |
| AI HL | Not a natural fit | The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level. |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: error estimates without the bound; and degrees vs radians. Say how each one affects your answer.
Use of mathematics (E)
SL: maclaurin series of sin x (HL; explain for SL); alternating series error bounds; trigonometric identities for angle reduction; counting operations to compare methods, each explained in your own words and carried out correctly.
How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Compare with a Padé or Chebyshev approximation (explain it) for the same number of operations.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.
See a complete IA, marked
Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete pure maths exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Maclaurin series of sin x (HL; explain for SL), alternating series error bounds, trigonometric identities for angle reduction and counting operations to compare methods.
Where can I get data for this IA?
No data needed; compare your results with a calculator. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
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