Problem Solving with the Sine Rule, Cosine Rule, and Area

Trigonometry · Lesson 3 (synthesis) · Pearson Edexcel International A-Level Pure Mathematics 1

Driving Question: You now own every tool for non-right-angled triangles — the sine rule, the cosine rule, the area formula. But a real problem never tells you which one to pick up first. So how do you decide where to start when nobody hands you the method?

🧠 The Knowing World

This lesson adds no new formula. Instead it builds the skill that exams really test: choosing the right tool, in the right order, from a kit you already own. Here is the whole kit on one page.

Tool	What it gives you	Reach for it when you know…
Sine rule a/sin A = b/sin B = c/sin C	a missing side or angle	a side and the angle opposite it, plus one more piece (AAS, ASA, or the SSA "ambiguous" case)
Cosine rule a² = b² + c² − 2bc cos A	a missing side (or, rearranged, a missing angle)	two sides and the angle between them (SAS), or all three sides (SSS)
Area ½ ab sin C	the area of the triangle	two sides and the angle between them

The single most useful question to ask yourself is: "Do I have a side paired with its opposite angle?" If yes, the sine rule is usually the quickest way in. If not — if your known angle is wedged between two known sides — the sine rule has nothing to grip, and the cosine rule is your starting tool. The diagram below turns that judgement into a flowchart.

A decision flowchart for choosing the first tool. "Opposite" sends you to the sine rule; "between" (included) sends you to the cosine rule and the area formula.

Often a problem needs more than one tool in sequence: the cosine rule unlocks a side, that side feeds the sine rule for an angle, and the area formula finishes the job. Recognising which tool comes next is the same judgement applied again and again.

Seed of Change (Pearson Edexcel IAL P1, §3.1): Use the sine rule and the cosine rule, including the ambiguous case of the sine rule; use the area of a triangle in the form ½ ab sin C — selecting and combining these in multi-step, unstructured problems.

🌍 The Human Story

For most of history, the only way to measure something you couldn't walk across — a river, a valley, the distance to a ship at sea — was to turn it into a triangle. Surveyors would pace out one side they could measure (a "baseline"), measure two angles with a theodolite, and let the sine rule hand them the rest. This is triangulation, and it mapped whole countries: the Great Trigonometrical Survey of India spent most of the 19th century chaining triangles across a subcontinent to measure it. The same idea, sped up and satellite-borne, is how your phone's GPS fixes your position right now.

But the tools alone don't solve anything. What surveyors and navigators actually relied on was a way of thinking when the path forward isn't obvious. In 1945 the mathematician George Pólya wrote it down as four plain steps in his book How to Solve It — and they fit a trigonometry problem perfectly:

Understand the problem. What do I actually know, and what am I asked to find? Draw the triangle. Mark on every length and angle. Bearings? Turn them into interior angles first.
Make a plan. Which tool reaches the unknown? If I can't get there in one step, what intermediate quantity would unlock it? (A side I could find with the cosine rule might be exactly what the sine rule then needs.)
Carry out the plan. Do the algebra carefully, keeping full accuracy until the final rounding.
Look back. Is the answer sensible? Is the biggest side opposite the biggest angle? Does the distance feel realistic for the picture?

The diagram-first habit in step 1 is not a nicety — it is where most of the thinking happens. A clear, labelled figure usually shows you which tool fits before you've written any algebra.

🎨 The Expressive Self

Pick one of these and make the reasoning your own — no full solution required, just the thinking:

Write a short "first move" rule in your own words — the test you'd run on any triangle problem to decide whether to start with the sine rule or the cosine rule. Try to make it one sentence a friend could memorise.
Sketch a triangle problem of your own where you'd have to use two tools in sequence. Don't solve it — just annotate the order: "cosine rule first, because…; then sine rule, because…".
Explain to someone (out loud, or in a voice note to yourself) why a known angle between two known sides rules out starting with the sine rule. If you can teach that one idea clearly, you understand the whole choice.

There's no single right expression here — the goal is to hear yourself reason about choosing, because that's the part an exam can't give you.

🛠️ The Designed World

Here is a real navigation problem. You'll solve it the way a navigator would — one decision at a time. Each step unlocks only when the previous one is right, and the chart fills in as you go, so you can always see what you know.

The situation. A ship leaves harbour H and sails 15 km on a bearing of 045° to point A. It then turns and sails 11 km on a bearing of 150° to point B. The skipper needs to know how far B is from harbour, what bearing to steer to return straight home, and the area of sea enclosed by the route.

Show the North-line guides at A and B (each extended above and below its point)

Step 1 — Understand: find the interior angle at A

A bearing is measured clockwise from North. Turn the two bearings at A into the actual interior angle ∠HAB of the triangle. (Give your answer in degrees.)

The bearing of A from H is 045°. Standing at A and looking back at H, you face the reverse bearing — add 180°. The bearing onward from A to B is 150°. The interior angle ∠HAB is the angle between those two directions at A.

Reverse bearing A→H = 045° + 180° = 225°. The direction A→B is 150°. The angle between them is 225° − 150°.

Step 2 — Plan & do: find the distance HB

You now know two sides (HA = 15, AB = 11) and the angle between them. Which tool fits? Find HB, to 1 decimal place (km).

Two sides and the included angle ⇒ cosine rule. With the angle at A opposite the side HB, write HB² = HA² + AB² − 2·HA·AB·cos A.

Substitute: HB² = 15² + 11² − 2(15)(11)cos 75°. Work out the right-hand side, then take the square root.

Step 3 — Next tool: find the angle ∠AHB

Now you have a side (HB) paired with its opposite angle (A) — so the sine rule has something to grip. Find ∠AHB, to 1 decimal place (degrees).

Sine rule: sin(∠AHB) / AB = sin A / HB. The side opposite ∠AHB is AB = 11.

Rearrange to sin(∠AHB) = AB·sin A / HB = 11·sin 75° / HB, then apply sin⁻¹. Use your unrounded HB for full accuracy.

Step 4 — Finish: find the area enclosed by the route

Find the area of triangle HAB, to 1 decimal place (km²).

You have two sides and the angle between them — that's exactly what the area formula ½ ab sin C wants.

Area = ½ · HA · AB · sin A = ½ · 15 · 11 · sin 75°.

Solved — and now the navigator's answer. The angle ∠AHB you found is measured from the H→A direction (bearing 045°). Since B lies clockwise of A as seen from H, the bearing of B from the harbour is 045° + ∠AHB ≈ 086°. To steer home from B, the skipper reverses it: 086° + 180° = 266°. So: B is about 16.1 km from harbour on a bearing of 086°, the route home is bearing 266°, and the triangle of sea enclosed is about 79.7 km².

Investigate with the chart:

Notice the order you were forced into: cosine rule before sine rule. Why can't you find ∠AHB first? (What would the sine rule need that you don't yet have?)
In Step 3 the sine rule gives sin(∠AHB) ≈ 0.658, which has two solutions between 0° and 180°. Why is the obtuse one impossible here? (Look at the triangle: which is the longest side, and which angle must therefore be largest?)
Suppose the skipper had instead measured HB directly and knew all three sides. Which tool would now be the natural first move, and why?
Re-do Step 4 using ½ · HA · HB · sin(∠AHB) instead of ½ · HA · AB · sin A. You should get the same area — why must you?

💛 The Living Body

The hardest moment in a problem like this is the very first one — the blank-page feeling of "I don't know where to start." That feeling is not a sign you can't do it. It's just the normal gap between reading a problem and seeing into it, and it closes the instant you do one small, concrete thing: draw the triangle and label what you know.

Notice how the workbench never asked you to see the whole solution at once. It asked for one decision at a time. That's a habit you can carry into any problem: you don't need the full route before you take the first step — you need only the next step.

Pause and check in with yourself:

Where did you feel stuck, and what got you moving again — the diagram, a hint, or just naming what you knew?
When the answer came out, did it feel right before you checked it? Building that sense of "this is about the right size" is real mathematical skill, not a lucky guess.
Next time the blank-page feeling shows up, what's the one small move you'll make first?

🛠 Final Task

Your task: become the problem-setter. Design one multi-step triangle problem of your own — set in a real context (navigation, surveying, a sports field, a roof truss, anything) — that genuinely needs at least two of the three tools used in sequence. Then write a full model solution that:

begins with a clearly labelled diagram;
states, at each step, which tool you're using and why it fits (name the configuration: SAS, SSS, AAS, …);
keeps full accuracy through the working and rounds only at the end;
finishes with a "look back" sentence checking the answer is sensible.

Swap with a partner if you can: solving each other's problems is the real test of whether the setup is clear and consistent.

Seed of Change (Pearson Edexcel IAL P1, §3.1): Select and combine the sine rule, cosine rule (including the ambiguous case), and the area formula ½ ab sin C to solve multi-step problems presented in unstructured, real-world form.

How your work will be assessed — the Haven Maths Rubric:

Strand	What we're looking for here
Conceptual Understanding	You choose tools by the configuration (opposite vs included), not by guesswork, and can justify each choice.
Fluency & Accuracy	Correct substitution and algebra; full accuracy carried through; sensible final rounding.
Application to Problems	You translate an unstructured, real-world situation (including bearings) into a correctly labelled triangle and a working plan.
Independence & Reflection	You design a consistent problem of your own and check that your answer is reasonable ("look back").

Ready to build fluency? The practice companion has exam-style questions with Socratic support whenever you get stuck.

✏️ Now Practise →