Cosine Rule Explained
The cosine rule helps you solve non-right triangles. Use it to find a missing side when you know two sides and the included angle, or to find an angle when you know all three sides.
Core formula: \(c^2 = a^2 + b^2 - 2ab\cos(C)\)
Quick answer: The cosine rule, also called the law of cosines, is used for triangles that are not necessarily right-angled. To find a side, use \(c^2 = a^2 + b^2 - 2ab\cos(C)\). To find an angle, rearrange it as \(\cos(C) = \frac{a^2+b^2-c^2}{2ab}\).
Cosine Rule
Use \(c^2 = a^2 + b^2 - 2ab\cos(C)\) when two sides and the included angle are known.
Find Missing Angles
Use \(\cos(C) = \frac{a^2+b^2-c^2}{2ab}\) when all three side lengths are known.
Complete Triangle Analysis
Calculate sides, angles, area, perimeter, medians, altitudes, and triangle type.
What is the cosine rule?
The cosine rule is a triangle formula used when Pythagoras' theorem is not enough. Pythagoras works only for right-angled triangles. The cosine rule works for any triangle, as long as you have the right combination of sides and angles.
To find a side:
\[c^2 = a^2 + b^2 - 2ab\cos(C)\]
To find an angle:
\[\cos(C)=\frac{a^2+b^2-c^2}{2ab}\]
In the formula, \(C\) is the angle between sides \(a\) and \(b\), and side \(c\) is opposite angle \(C\). This opposite-side/opposite-angle relationship is the most important naming rule to remember.
Cosine rule triangle layout
A triangle showing sides a and b meeting at angle C, with side c opposite angle C.
When should you use the cosine rule?
- Use it for SAS: two sides and the included angle are known, and you need the third side.
- Use it for SSS: all three sides are known, and you need an angle.
- Use it when the triangle is not right-angled or when Pythagoras' theorem does not apply.
Cosine Rule Calculator
Choose a mode based on what your triangle question gives you. The first two options cover the most common exam-style cosine rule problems.
Find Missing Side (Two Sides + Included Angle)
Cosine rule worked examples
Example 1: Find a missing side
Problem: A triangle has sides \(a = 7\), \(b = 10\), and included angle \(C = 60^\circ\). Find side \(c\).
Solution:
\[c^2 = 7^2 + 10^2 - 2(7)(10)\cos(60^\circ)\]
\[c^2 = 49 + 100 - 140(0.5)=79\]
\[c = \sqrt{79} \approx 8.89\]
Answer: \(c \approx 8.89\).
Example 2: Find a missing angle
Problem: A triangle has sides \(a = 5\), \(b = 8\), and \(c = 10\). Find angle \(C\), opposite side \(c\).
Solution:
\[\cos(C)=\frac{5^2+8^2-10^2}{2(5)(8)}\]
\[\cos(C)=\frac{25+64-100}{80}=-0.1375\]
\[C=\cos^{-1}(-0.1375)\approx 97.9^\circ\]
Answer: \(C \approx 97.9^\circ\).
Common mistakes to avoid
- Do not use Pythagoras unless the triangle is right-angled.
- Make sure the angle used in the side formula is the included angle between the two known sides.
- When finding an angle, use the side opposite that angle as the subtracted side in the numerator.
- Check calculator mode: degrees and radians give different input values.
Triangle Formulas Reference
Triangle Classification
By Angles
Acute: all angles < 90°
Right: one angle = 90°
Obtuse: one angle > 90°
By Sides
Equilateral: all sides equal
Isosceles: two sides equal
Scalene: all sides different
Checks
Angle sum: always 180°
Triangle inequality: a + b > c
Longest side: opposite largest angle
Cosine rule FAQs
What is the cosine rule?
The cosine rule is a formula for solving non-right triangles. The common side formula is \(c^2 = a^2 + b^2 - 2ab\cos(C)\), where \(C\) is the angle between sides \(a\) and \(b\).
When do you use the cosine rule?
Use the cosine rule when you know two sides and the included angle and need the third side, or when you know all three sides and need an angle.
Is the cosine rule the same as the law of cosines?
Yes. “Cosine rule” and “law of cosines” refer to the same triangle formula. “Cosine rule” is common in UK-style curricula, while “law of cosines” is common in US-style curricula.
How do you find an angle using the cosine rule?
Rearrange the formula: \(\cos(C)=\frac{a^2+b^2-c^2}{2ab}\). Then use inverse cosine: \(C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)\).
Can the cosine rule be used for right triangles?
Yes, but Pythagoras is usually simpler. If \(C=90^\circ\), then \(\cos(90^\circ)=0\), so the cosine rule becomes \(c^2=a^2+b^2\).
What is the difference between sine rule and cosine rule?
The cosine rule is best for SAS and SSS information. The sine rule is best when you know a matching side-angle pair, such as ASA or AAS cases.
Related RevisionTown resources
Use these pages to strengthen the wider trigonometry and exam-revision cluster.
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
