The Cosine Rule: GCSE Maths Formula, Examples & When to Use It

The cosine rule is one of the most important trigonometry formulas on the GCSE Maths higher tier. This guide is part of our complete GCSE Maths revision guide — start there for an overview of all topics and revision strategies. It lets you find missing sides and angles in any triangle, not just right-angled ones. If you’ve been relying on Pythagoras and SOHCAHTOA, the cosine rule opens up a whole new category of problems.

This guide covers the formula, when to use it, worked examples, and the mistakes that cost students marks every year.

The Cosine Rule Formula

The cosine rule has two forms, depending on whether you’re finding a side or an angle.

Finding a Side

a² = b² + c² - 2bc cos(A)

Here, a is the side you want to find, and A is the angle opposite that side. The sides b and c are the other two sides of the triangle.

Finding an Angle (Rearranged Form)

cos(A) = (b² + c² - a²) / 2bc

This version lets you find an angle when you know all three sides.

Notice that when angle A is 90 degrees, cos(A) = 0, and the cosine rule simplifies to a² = b² + c². That’s just Pythagoras’ theorem. The cosine rule is essentially a generalisation of Pythagoras that works for all triangles.

When to Use the Cosine Rule

You need the cosine rule in two specific situations:

Situation 1: Two sides and the included angle (SAS). You know two sides and the angle between them, and you want to find the third side. The key word is “included” — the angle must be the one enclosed between the two known sides.

Situation 2: Three sides (SSS). You know all three sides and want to find an angle. Use the rearranged form.

If you don’t have one of these setups, you probably need the sine rule instead (which works when you have a side and its opposite angle).

A huge proportion of dropped marks on this topic comes from students jumping straight into a formula without checking they’ve got the right setup — then wondering why their answer makes no sense. Spend the extra ten seconds on the decision before you reach for the formula.

Worked Examples

Example 1: Finding a Missing Side

In triangle ABC, side b = 7 cm, side c = 10 cm, and angle A = 65 degrees. Find side a.

Solution:

a² = b² + c² - 2bc cos(A)

a² = 7² + 10² - 2(7)(10) cos(65)

a² = 49 + 100 - 140 cos(65)

a² = 149 - 140(0.4226)

a² = 149 - 59.16

a² = 89.84

a = 9.48 cm (to 3 significant figures)

Example 2: Finding a Missing Angle

In triangle PQR, side p = 8 cm, side q = 11 cm, and side r = 14 cm. Find angle R.

Solution: Using the rearranged form. R is the angle we want, and r is the side opposite it (14 cm). The other two sides are p = 8 and q = 11.

cos(R) = (p² + q² - r²) / 2pq

cos(R) = (64 + 121 - 196) / 2(8)(11)

cos(R) = (185 - 196) / 176

cos(R) = -11 / 176

cos(R) = -0.0625

R = cos⁻¹(-0.0625)

R = 93.6 degrees (to 1 decimal place)

Notice that cos(R) is negative. This tells us the angle is obtuse (greater than 90 degrees). Perfectly normal. This is one of the advantages of the cosine rule: it handles obtuse angles without any extra work.

Example 3: A Two-Step Problem

In triangle XYZ, side x = 6 cm, side y = 9 cm, and angle Z = 48 degrees. Find side z, then use the sine rule to find angle X.

Step 1: Find side z using the cosine rule.

z² = x² + y² - 2xy cos(Z)

z² = 36 + 81 - 2(6)(9) cos(48)

z² = 117 - 108(0.6691)

z² = 117 - 72.26

z² = 44.74

z = 6.69 cm (to 3 significant figures)

Step 2: Find angle X using the sine rule.

sin(X) / x = sin(Z) / z

sin(X) / 6 = sin(48) / 6.69

sin(X) = 6 sin(48) / 6.69

sin(X) = 6(0.7431) / 6.69

sin(X) = 0.6665

X = sin⁻¹(0.6665)

X = 41.8 degrees (to 1 decimal place)

This shows how the cosine rule and sine rule often work together in multi-step problems.

Cosine Rule vs Sine Rule: When to Use Which

This decision trips up more students than any calculation error. Here’s a clear guide:

Use the cosine rule when you have:

• Two sides and the included angle (the angle between them) and want the third side
• All three sides and want an angle

Use the sine rule when you have:

• A side and its opposite angle, plus one other side or angle
• Two angles and one side (find the third angle first since they sum to 180)

A quick test: if you can pair up a side with its opposite angle, use the sine rule. If you can’t, use the cosine rule.

Common Mistakes

A near-universal mock-paper error: using whichever angle the question gives, rather than checking it’s the angle opposite the side you want. If you’re finding side a, you must use angle A — the angle at vertex A, which sits opposite side a. The pattern is always the same: correct formula, correct substitution method, completely wrong angle. Zero marks for the final answer.

Calculator in the Wrong Mode

Your calculator must be in degrees mode, not radians. If you get an answer that looks wildly wrong (say, a side length of 150 cm in a triangle with sides of 7 and 10), check your calculator mode immediately. Look for a “D” or “DEG” indicator on the display.

Sign Errors with Negative Cosine Values

When the angle is obtuse (greater than 90 degrees), cos(A) is negative. The “-2bc cos(A)” term actually becomes positive (negative times negative). Students often lose the negative sign during their working. Write out each step carefully.

Rounding Too Early

Don’t round intermediate values. Keep the full calculator display until your final answer, then round as instructed (usually 3 significant figures or 1 decimal place). Rounding partway through introduces errors that accumulate, especially in multi-step problems.

A simple rule that prevents most premature-rounding errors: leave every number on the calculator until the very last line. Only round once, at the end.

Confusing Labelling

Lowercase = side. Uppercase = angle at the opposite vertex. Side a is opposite angle A. Side b is opposite angle B. If the triangle uses different letters (like PQR), adjust: side p is opposite angle P.

Exam Technique

Show your substitution step. After writing the formula, show the numbers substituted in. This earns you a method mark even if you make an arithmetic error later.

State the formula you’re using. Writing “cosine rule: a² = b² + c² - 2bc cos(A)” at the start makes your method clear and earns credit.

Draw and label the triangle. If the question doesn’t provide a diagram, sketch one. Label all sides and angles you know. This makes it much easier to spot which formula version to use.

Check your answer is reasonable. The longest side is always opposite the largest angle. If your calculated side is longer than both given sides but the angle between them is less than 60 degrees, something’s gone wrong.

Beyond GCSE

At A-Level, the cosine rule appears in pure mathematics and mechanics. You’ll use it in vector problems, navigation bearings, and force diagrams. The formula stays the same, but contexts become more varied. A strong grasp now sets you up well.

The cosine rule also underpins the dot product of vectors, which you’ll meet in Further Maths or Physics at A-Level.

Related GCSE Maths Guides

• Pythagoras’ theorem guide — for right-angled triangles and 3D applications
• Geometry and measures guide — angles, shapes, transformations, and trigonometry
• GCSE Maths formula sheet guide — the sine and cosine rules are on the sheet; know when to use each

Ready to practise? Try GCSE Maths questions on UpGrades to find and fix your weak spots.

How to Use This Guide

Read through the worked examples with a calculator in hand — actually punch in the numbers yourself. Then try a few past paper questions (AQA and Edexcel both have plenty). If you’re getting stuck on deciding which rule to use, go back to the “Cosine Rule vs Sine Rule” section and quiz yourself on random triangle setups. That decision-making skill is half the battle. Once it clicks, the calculations are the easy part.