IA idea · Trigonometry, navigation & surveying

What is the best angle to sail upwind?

AA HLAA SLAI HL Ambitious Also in: Optimisation, Kinematics

Research question

A boat can't sail straight into the wind, so it zig-zags. Given a boat's speed at each angle to the wind, which heading gets it upwind fastest, and how many tacks should it make on a real course?

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

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Why it makes a good exploration

It is an optimisation that uses trigonometry for the useful component of velocity. With a published speed table for a real boat class, the answer can be checked against what sailors do.

The mathematics you'll need

Velocity components: v cos θ towards the wind
Fitting a function to speed against angle
Maximising the upwind component with calculus
Course length and time for different numbers of tacks
Time lost per tack as a parameter

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

Find a published speed table ('polar diagram') for a dinghy or yacht class and cite it, or ask a sailing club for one.

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

Explain velocity made good towards the wind.
Fit a model to the speed table.
Find the optimal angle with calculus.
Model a course with tacks and find the best number.
Compare with sailors' practice and reflect.

Pitfalls that cost marks

Optimising speed instead of upwind progress.
Fitting a model that misbehaves between data points.
Ignoring the time lost tacking.

Showing personal engagement

Use a boat you sail or have seen.
Ask a sailor how they choose the angle.
Find how the best angle changes with wind strength.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Velocity components: v cos θ towards the wind; Fitting a function to speed against angle
AA HL	Good fit	Velocity components: v cos θ towards the wind; Fitting a function to speed against angle
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	Good fit	Velocity components: v cos θ towards the wind; Fitting a function to speed against angle

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Take the measurements yourself, choose the place, and compare at least two methods. Explaining why you chose each position or angle is engagement an examiner can see.

Reflection (D)

Analyse measurement error: how much does a small error in an angle change the answer, which method is most robust, and would you trust the result for the decision you are making? For this idea, start with: optimising speed instead of upwind progress — say how it affects your answer.

Use of mathematics (E)

SL: Right-angled trigonometry, sine and cosine rules, bearings and 3D trigonometry applied correctly to your own measurements, with an error analysis.

HL: Error propagation with derivatives, vectors in three dimensions, spherical geometry derived rather than quoted, or an optimisation of where to measure from.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Add a tidal current as a vector and find the new best headings.

Extending it for HL

Use derivatives to find how sensitive the result is to each measured angle, and choose the measuring position that minimises the error.

See a complete IA, marked

Our annotated exemplar How long does a skydiver take to reach terminal velocity? (AA HL) asks a different question, but shows how a complete trig & navigation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).