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 have 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 are 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 of the triangle.

Notice that when the angle A is 90 degrees, cos(A) = 0, and the cosine rule simplifies to a² = b² + c², which is 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 of the triangle and you want to find an angle. Use the rearranged form.

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

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. Here, 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, which tells us the angle is obtuse (greater than 90 degrees). This is perfectly normal and 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 example 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 is one of the most important decisions in GCSE trigonometry. Here is 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 (you can find the third angle first since angles sum to 180)

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

Common Mistakes

Using the Wrong Angle

The angle in the cosine rule must be opposite the side you are finding. If you are calculating side a, you must use angle A (the angle at vertex A, which is opposite side a). Using the wrong angle is the most common error and will give a completely wrong answer.

Calculator in the Wrong Mode

Your calculator must be in degrees mode, not radians. If you get an answer that looks wildly wrong (for example, a side length of 150 cm in a triangle with sides of 7 and 10), check your calculator mode immediately. On most scientific calculators, 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. This means the “-2bc cos(A)” term actually becomes positive (negative times negative). Students sometimes lose the negative sign during their working. Write out each step carefully and keep track of signs.

Rounding Too Early

Do not round intermediate values during your working. Keep the full calculator display until you reach your final answer, then round as the question instructs (usually 3 significant figures or 1 decimal place). Rounding partway through introduces errors that accumulate, especially in multi-step problems.

Confusing Labelling

In the cosine rule, the lowercase letter always refers to the side, and the uppercase letter refers to the 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 accordingly: 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 are using. Writing “cosine rule: a² = b² + c² - 2bc cos(A)” at the start of your answer makes your method clear and earns credit.

Draw and label the triangle. If the question does not provide a diagram, sketch one yourself. Label all the sides and angles you know. This makes it much easier to identify which version of the formula to use and reduces the chance of mixing up sides and angles.

Check your answer is reasonable. In any triangle, 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 has gone wrong.

Beyond GCSE

At A-Level, the cosine rule appears in both pure mathematics and mechanics. It is used in vector problems, navigation bearings, and force diagrams. The formula itself stays the same, but the contexts become more varied. A strong grasp of when and how to apply it at GCSE sets you up well for further study.

The cosine rule is also the foundation for understanding the dot product of vectors, which you will meet if you study 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

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