What is the Cosine Rule? (With Explanation & Real Examples)

The Cosine Rule, also known as the Law of Cosines, is a powerful trigonometric identity used to find the unknown side or angle in any triangle—not just right-angled ones. It becomes especially useful when you have either:

Two sides and an included angle (SAS), or
All three sides of a triangle (SSS)

Formula of the Cosine Rule

$c^{2} = a^{2} + b^{2} - 2 a b \cos (C)$

Where:

a, b, and c are the lengths of the triangle sides
C is the angle opposite side c

Cosine Rule Calculator

Calculation Type:

Side a:

Side b:

Angle C (degrees):

Enter values and click Calculate to see the result

Cosine Rule:
• To find a side (c): $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$
• To find an angle (C): $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

the proof that there are only five equable triangles (integer sided) was done in 1904

this version of a proof is largely due to David Wells with small bits of help from me

it is complex but involves only GCSE tools

roof truss

one roof making company calls the various truss designs:
Howe truss
Fink truss
Attic truss

which one uses most wood?

quite long and complex calculations involving the sine and cosine rule

symmetry is assumed

the Fink truss does not have struts that are perpendicular to the main, 6m lengths (even though it looks like they might be)

cosine rule

a powerpoint is here

the ambiguous case, where you are given angle, side, side (condition: A.S.S.) can lead to two integer solutions, obtained by factorising and solving a quadratic equation

Why Does It Matter?

The cosine rule is foundational in GCSE Maths, IB, and A-Level syllabi. It helps you solve triangle problems that can’t be handled by Pythagoras or sine rules alone. It’s also used in navigation, physics, engineering, and even machine learning models involving geometry.

Cosine Rule Derivation Using Geometry Tools (Proof)

Did you know? The proof that there are only five equable integer-sided triangles was first presented in 1904.

This article presents a simplified version of the proof based on work by David Wells, adapted for GCSE-level learners. It involves:

Triangle construction
In-circle radius analysis
Perimeter-to-area comparison

These diagrams below illustrate how cosine values and side lengths interact across different triangle configurations.

Worked Example (Step-by-Step)

Question:
Given triangle ABC where AB = 9 cm, BC = 7 cm, and angle B = 100°, find side AC.

Solution:
Using the formula:

$c^2 = 9^2 + 7^2 – 2(9)(7)\cos(100^\circ)$ $c^{2} = 81 + 49 - 126 \cdot \cos (100^{\circ})$

Use a calculator to solve.

Past Paper Practice Questions

We’ve included curated past paper problems using cosine rule logic. Each includes:

Difficulty level (easy, moderate, hard)
Complete solutions
Common mistakes to avoid

Tips From Top Educators

Always check if it’s non-right triangle before using cosine rule
Use exact values for cosine (avoid rounding early)
Compare with sine rule to confirm your result

FAQs – Cosine Rule Study Help

Q1: Can I use cosine rule on a right triangle?
Yes, but it’s not necessary—Pythagoras is simpler.

Q2: What’s the difference between the sine and cosine rule?
Sine rule is for angles opposite known sides, cosine rule is for enclosed angles.

Q3: Is the cosine rule in the GCSE formula sheet?
Yes, it’s provided. Make sure you know when to use it.