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

Advanced Cosine Rule Calculator

Complete Triangle Analysis & Solving Tool

📐 Cosine Rule

c² = a² + b² - 2ab cos(C). Essential for solving any triangle when you know two sides and the included angle.

📊 Triangle Properties

Calculate angles, sides, area, perimeter, circumradius, inradius, and more triangle properties.

🔍 Complete Analysis

Get comprehensive triangle analysis including type classification, medians, altitudes, and geometric centers.

Select Calculation Mode:

Find Missing Side (Two Sides + Included Angle)

Side a: Length of the first known side

Side b: Length of the second known side

Included Angle C (between sides a and b):

Angle between the two known sides (0° < C < 180°)

📐 Triangle Formulas Reference

Formula Type	Equation	Use Case
Cosine Rule (Side)	c² = a² + b² - 2ab cos(C)	Find a side when 2 sides and included angle known (SAS)
Cosine Rule (Angle)	cos(C) = (a² + b² - c²) / (2ab)	Find an angle when all 3 sides known (SSS)
Sine Rule	a/sin(A) = b/sin(B) = c/sin(C)	Find sides or angles in ASA, AAS, SSA cases
Heron's Formula	Area = √[s(s-a)(s-b)(s-c)], s = (a+b+c)/2	Calculate area from three sides
Area (SAS)	Area = (1/2)ab sin(C)	Calculate area from two sides and included angle
Circumradius	R = abc / (4×Area)	Radius of circumscribed circle
Inradius	r = Area / s	Radius of inscribed circle
Altitude	h = (2×Area) / base	Height from vertex to opposite side

📊 Triangle Classification

By Angles

Acute: All angles < 90°

Right: One angle = 90°

Obtuse: One angle > 90°

By Sides

Equilateral: All sides equal (all angles 60°)

Isosceles: Two sides equal

Scalene: All sides different

Special Properties

Sum of angles: Always 180°

Triangle inequality: a + b > c

Largest angle: Opposite longest side

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.