Cosine Rule (AA SL)
When a triangle gives you two sides and the angle trapped between them, or all three sides and nothing else, the sine rule has no angle-side pair to work with - that's exactly when the cosine rule takes over. This page covers both directions of the formula, when each applies, and the sign error that quietly wrecks more solutions than any other. It's part of the broader Triangles (Sine/Cosine Rules) topic.
16 questions on this sub-topic.
The cosine rule
Covered under IB syllabus reference SL3.2: the cosine rule, in both its side-finding and angle-finding forms. Both versions are given in the formula booklet, so recognising SAS or SSS data is the real skill.
Finding a side (SAS)
\(c^2=a^2+b^2-2ab\cos C\)
Use when you know two sides and the included angle between them. Take a square root at the end - a side length is always positive.
Finding an angle (SSS)
\(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\)
The rearranged form - use when all three sides are known and nothing else. A negative result means the angle is obtuse.
Given an angle-side pair instead? See Sine rule - or check the full topic page for GDC guidance.
Worked examples
In triangle \(ABC\), \(b = 7\) cm, \(c = 10\) cm and \(A = 55^\circ\).
Find side \(a\).
Worked solution
Two sides and the included angle \(A\) are known, so use the cosine rule: \(a^2=b^2+c^2-2bc\cos A.\) M1
Substitute: \(a^2=49+100-140\cos 55^\circ\approx 68.7.\) A1
Square root: \(a\approx 8.29\) cm. A1
A triangle has sides 5 cm, 7 cm and 9 cm.
Find the size of the largest angle.
Worked solution
The largest angle faces the longest side (9 cm), so use \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}=\dfrac{25+49-81}{70}=-0.1.\) M1
\(C=\cos^{-1}(-0.1)\approx 95.7^\circ.\) A1
In quadrilateral \(ABCD\), \(AB = 10\), \(BC = 7\), angle \(ABC = 80^\circ\). Diagonal \(AC\) is found, then \(CD = 9\) and angle \(ACD = 40^\circ\).
(a) Find \(AC\).
(b) Find \(AD\).
Worked solution
(a) Find \(AC\). Cosine rule in \(\triangle ABC\): \(AC^2=100+49-140\cos 80^\circ\approx 124.7\Rightarrow AC\) M1
\(\approx 11.2.\) A1
(b) Find \(AD\). Cosine rule in \(\triangle ACD\): \(AD^2=11.2^2+9^2-2(11.2)(9)\cos 40^\circ\approx 51.4\Rightarrow AD\approx 7.17.\) M1A1
Common mistakes
- Reaching for the sine rule with SAS data. If you only know two sides and the angle between them, there's no angle-side pair to plug into the sine rule - you need the cosine rule instead.
- Sign errors in the cosine rule. Writing \(c^2=a^2+b^2+2ab\cos C\) instead of \(-2ab\cos C\) - the minus sign is what makes the formula reduce to Pythagoras when \(C=90^\circ\).
- Forgetting a negative cosine still gives a valid angle. A negative value for \(\cos C\) just means \(C\) is obtuse - don't dismiss it as an arithmetic mistake and go hunting for a "nicer" positive answer.
Ready to practise properly?
16 cosine-rule questions, marked instantly like the real exam.
Quick answers
When do I use the cosine rule?
Use the cosine rule when you know two sides and the angle between them (SAS), to find the third side, or when you know all three sides (SSS), to find any angle.
What is the cosine rule formula?
\(c^2=a^2+b^2-2ab\cos C\) to find a side, or \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\), its rearranged form, to find an angle.