The Cosine Rule

The Cosine Rule
When Pythagoras needs a correction — Pure 1, Trigonometry (Lesson 2)

Driving Question: Pythagoras's theorem works perfectly — for right-angled triangles. But what about all the other triangles? If the angle isn't 90°, by how much does a² + b² miss c² — and can we measure that miss with a single correction term?

In Lesson 1 you met the sine rule: a beautiful proportion that handles triangles when you have a side and its opposite angle. But the sine rule has a blind spot — it needs that side-and-opposite-angle pair to start. What if you have two sides and the angle between them? Or all three sides and no angles? The sine rule can't get going. For these cases, we need a different rule — and it turns out to be one of the most elegant generalisations in mathematics: Pythagoras's theorem with a correction term that vanishes precisely when the angle is 90°. Open the panels in any order, and watch the proof animation in the Knowing World — it's a beautiful payoff for a single perpendicular dropped from a single vertex.

🧠 The Knowing World

The convention

The same labelling convention from Lesson 1: in triangle ABC, vertices are A, B, C, and each side is named after the angle opposite it — so side a is opposite vertex A, and so on.

The same convention: each side carries the lowercase letter of the angle opposite to it.

For any triangle ABC:

c² = a² + b² − 2ab cos C

By symmetry, there are three versions — one for each side, with the included angle (the angle between the two sides on the right) at the end:

a² = b² + c² − 2bc cos A b² = a² + c² − 2ac cos B c² = a² + b² − 2ab cos C

The beautiful insight: it's Pythagoras with a correction

Look at the formula c² = a² + b² − 2ab cos C and try the special case C = 90°. Since cos 90° = 0, the correction term −2ab cos C vanishes, and we're left with:

c² = a² + b²

...which is just Pythagoras's theorem. So the cosine rule is Pythagoras — extended to triangles that aren't quite right-angled, with a correction term that measures exactly how far off from a right angle we are.

When C < 90° (acute), cos C > 0, so the correction −2ab cos C is negative, making c² smaller than a² + b². The opposite side is shorter than Pythagoras would predict — because the two sides are leaning towards each other.
When C > 90° (obtuse), cos C < 0, so the correction is positive, making c² larger than a² + b². The opposite side is longer — because the two sides are splayed apart.
And when C = 90°, the correction is exactly zero. Pythagoras lives at this single special angle.

This is one of those moments in mathematics where a formula stops being a thing to memorise and starts feeling like a thing that had to be true.

Why does it work?

The proof is structurally similar to the sine rule's — drop a perpendicular and use what you know — but this time the tool is Pythagoras rather than basic trig ratios. Click Next step to build the proof.

Click Next step to begin. We'll drop a perpendicular, then apply Pythagoras to one of the right-triangles — and the cosine rule will fall out.

Step 0 of 6

The cosine rule proof (printable summary). In triangle ABC, drop a perpendicular from B to side AC; call its foot D, its length h, and let AD = x. Then DC = b − x. In the LEFT right-triangle ABD: cos A = x/c (so x = c cos A) and h = c sin A. In the RIGHT right-triangle BDC: by Pythagoras, a² = h² + (b − x)². Substituting h = c sin A and x = c cos A: a² = (c sin A)² + (b − c cos A)² = c² sin²A + b² − 2bc cos A + c² cos²A = c²(sin²A + cos²A) + b² − 2bc cos A = c² + b² − 2bc cos A. By relabelling, every version follows.

The proof above uses an acute triangle so the perpendicular falls inside; the result also holds for obtuse triangles, with a slightly different diagram but the same algebra after sign-handling.

When to reach for it

The cosine rule is the right tool for exactly the cases the sine rule can't start with:

SAS (two sides and the included angle): use c² = a² + b² − 2ab cos C directly to find the third side.
SSS (three sides, no angles): rearrange to find any angle. Rearranging c² = a² + b² − 2ab cos C for cos C:

cos C = (a² + b² − c²) / (2ab)

And similarly for cos A and cos B. Once you have cos C, take the inverse cosine to get C.

Worked example (SAS). A triangle has sides a = 7 and b = 9, with angle C = 50° between them. Find c.
• Apply the cosine rule: c² = 7² + 9² − 2(7)(9) cos 50° = 49 + 81 − 126 cos 50°.
• Compute: c² ≈ 130 − 126 × 0.6428 ≈ 130 − 80.99 ≈ 49.01.
• Take the square root: c ≈ 7.00.

Worked example (SSS). A triangle has sides a = 6, b = 8, c = 11. Find angle C.
• Rearranged cosine rule: cos C = (6² + 8² − 11²) / (2 × 6 × 8) = (36 + 64 − 121) / 96 = −21 / 96 ≈ −0.21875.
• Inverse cosine: C ≈ 102.6°. (The negative value of cos C immediately tells us C is obtuse — useful sanity check.)

🌱 Seeds of Change (Pearson Edexcel IAL Pure Mathematics 1, section 3.1): Use the cosine rule, in the forms a² = b² + c² − 2bc cos A (etc.) and the rearranged form cos A = (b² + c² − a²) / (2bc), to find missing sides and angles.

🌍 The Human Story

The cosine rule has a long pre-modern history — and a particularly elegant one, because it was discovered in stages by people who didn't yet have algebra in the modern sense.

Euclid had it, geometrically, around 300 BCE. In The Elements, Book II, Propositions 12 and 13, Euclid proves two results that — when you translate them into modern notation — are exactly the cosine rule for obtuse and acute triangles respectively. He doesn't use cosines (the concept didn't exist yet); he speaks of squares on sides, and rectangles of certain lengths. But the relationship is there, two thousand three hundred years old.
Al-Kashi gave it the modern shape in 1427. The Persian mathematician Jamshid al-Kashi, working in Samarkand, wrote down what is essentially the cosine rule in its modern algebraic form, using a trigonometric function instead of geometric squares. In France, the cosine rule is sometimes still called le théorème d'Al-Kashi.
It was renamed for cosines in the 17th century, when European mathematicians began using the cosine function explicitly. By then the result had been independently known to so many people that no one needed credit — it was just a useful fact.

The thinking point worth holding onto here: the cosine rule existed in three different forms in three different mathematical cultures, separated by centuries. Euclid's was geometric, al-Kashi's was algebraic-trigonometric, and the modern version is the algebraic shorthand we use now. Same truth, three languages. This is genuinely common in mathematics — the same idea found independently because it answers a real question that any culture doing geometry will eventually ask. The mathematics is, in a sense, waiting to be found.

🎨 The Expressive Self

Make the idea your own — pick whichever appeals. You only need to do one.

Prove it yourself. Sketch any triangle, drop a perpendicular from one vertex, and reconstruct the cosine rule proof in your own hand. The action of drawing the perpendicular yourself — and watching the algebra fall out — locks in the result far better than reading.

The correction story. In your own words (a paragraph, or a drawing with annotations), explain what the −2ab cos C term does. Pick three angles — say 30°, 90°, and 150° — and describe how the correction changes between them. You've understood it when you can explain why the formula reaches Pythagoras at exactly one angle.

Build an obtuse-triangle problem. Make up a triangle with sides 7, 9, and some unknown side c, and an angle between the 7 and 9 that's obtuse — say 110°. Compute c using the cosine rule. Then change the angle to 70° (the supplementary angle) and compute c again. The two answers should bracket what you'd get at exactly 90°. Sketch all three triangles.

From the answer backwards. Take a triangle with sides 5, 7, 8. Without computing first, predict whether each angle is acute or obtuse, and why. Then use the rearranged cosine rule to check. Were your predictions right? What helped you guess well?

🛠️ The Designed World

Two interactives here. The first verifies the cosine rule on a triangle you control — watch c² agree with a² + b² − 2ab cos C as you change the shape. The second is a Pythagoras Comparator: fix two sides, then sweep angle C from 0° to 180° and watch the correction term breathe — vanishing at 90°, becoming negative below it, positive above it.

Side a: 7

Side b: 9

Angle C (between a and b): 50°

Investigate with the Verifier (mode 1):

Set a = 7, b = 9, C = 50° (the default). Read off c² from the formula. Check by hand: 49 + 81 − 126 cos 50°. Do you get the same value the readout shows?
Now set C = 90°. What's the value of cos C? What does the correction term −2ab cos C become? What does c² equal?
Set C = 60° with a = b = 6. Predict c before reading the answer. (Hint: this is an equilateral situation.)

Investigate with the Pythagoras Comparator (mode 2):

Fix a = 5, b = 7 and slowly drag C from 10° to 170°. Watch the correction term. At what angle does it cross zero? What's special about that moment?
At C = 90°, what does c equal? Verify that c² = 25 + 49 = 74, so c = √74 ≈ 8.60.
Now set C = 30° and then C = 150° (the supplementary angle). How do the two values of c compare? Why does that make geometric sense?
Predict before checking: if C = 175°, what's c approximately equal to? (Hint: think about the triangle nearly flattening out.)

The deep insight that makes the cosine rule so beautiful: Pythagoras's theorem isn't a special case added on top of the cosine rule — Pythagoras is what the cosine rule becomes at one particular angle. The right-angled triangle is no longer a privileged shape that needs its own theorem. It's just the special case where the correction term happens to vanish.

💛 The Living Body

The cosine rule formula looks heavier than the sine rule on the page — three terms with squares and a cosine — and that visual weight can trigger a small panic before you've even started. A few things that help:

Anchor it to Pythagoras. Whenever you write the formula down, mentally check: "if C were 90°, this would just be Pythagoras." That anchor turns an alien-looking formula into a familiar one with a twist.
The included angle. The formula c² = a² + b² − 2ab cos C uses the angle between the two sides on the right (here, C is between a and b because a sits opposite A, b sits opposite B, and they meet at C). Getting the wrong angle is the most common source of marks lost — pause and identify which angle goes between which two sides every time.
The sign tells the story. When you find cos C via the rearranged formula, the sign tells you immediately whether the angle is acute or obtuse — negative cosine ⇔ obtuse angle. Use this as a sanity check before taking the inverse cosine.

And a small note on tools: scientific calculators reach the right answer only if they're in degrees mode (or radians mode, if your problem is in radians — but we'll meet radians properly in Lesson 4). Setting calculator mode is not optional. If your answer is wildly off from your sanity check, the calculator mode is the first thing to check.

Two reflections to sit with:

When you look at the formula c² = a² + b² − 2ab cos C, do you see "Pythagoras with a correction," or do you see "three squares and a cosine"? Either is fine — but the first reading is the one that makes the formula make sense.
Which felt cleaner to you, the sine rule or the cosine rule? The "right" answer is probably whichever you encountered first — but knowing your preference tells you which rule to default to when both could work.

🛠 Final Task

Build a "Cosine Rule Field Guide." Create a one- or two-page guide (handwritten and photographed, typed, or a slide) that teaches the cosine rule to a learner who has just finished Lesson 1 (the sine rule). It must contain:

The labelling convention (a labelled diagram) and a one-sentence reminder of why labelling correctly matters.
The cosine rule in its three side-finding forms and the rearranged angle-finding form, clearly stated.
An explanation of "Pythagoras with a correction" — in your own words (a paragraph, or annotated formula). Make it convincing.
A worked SAS problem — pick two sides and an included angle, find the third side. Show every step.
A worked SSS problem — pick three sides, find one of the angles using the rearranged formula. Show every step.
A side-by-side "sine rule vs cosine rule" — a small table or paragraph explaining when to reach for each. The decision criterion is essentially "do I have a side-and-opposite-angle pair?"
A short reflection (2–3 sentences): which part of this lesson felt most satisfying — the proof, the connection to Pythagoras, or the "when to use which" decision — and why.

🌱 Seeds of Change assessed here (Pearson Edexcel IAL Pure Mathematics 1, section 3.1):

Use the cosine rule in its side-finding form to find a missing side in a triangle (SAS case).
Use the rearranged cosine rule to find a missing angle (SSS case).

How this is assessed — Haven Maths Rubric

Strand	What we're looking for in this task
Conceptual Understanding	The "Pythagoras with a correction" insight is explained genuinely, not just stated. The included-angle convention is correct. The decision between sine rule and cosine rule is defensible.
Fluency & Accuracy	Calculations are correct end-to-end (including the right calculator mode), squares and square roots handled cleanly, angles in the right units.
Application to Problems	The worked SAS and SSS problems are the learner's own and are carried through to a clean final answer.
Independence & Reflection	The Field Guide is clear and self-made, the comparison with the sine rule is honest, and the reflection names a specific moment of insight.

Choose whatever form suits you — the mathematics is what's assessed; the presentation is yours.

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

✏️ Now Practise →

Opens the practice companion. Keep both files in the same folder. (If your browser blocks a new tab, it will open in this one — use Back to return.)