How does a calculator work out sin(x)?

AA HLAA SL Solid Also in: Calculus

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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

Compare with a Padé or Chebyshev approximation (explain it) for the same number of operations.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).