How to Use the Cosine Rule
The cosine rule lets you find a missing side or angle in any triangle, not just right-angled ones, and it is a core Edexcel IGCSE Maths skill on the Higher paper. You reach for it in the two situations the sine rule cannot handle: when you know two sides and the angle between them (to find the third side), or when you know all three sides (to find any angle). This page shows you how to use the cosine rule step by step, with the side formula a squared equals b squared plus c squared minus 2bc cos A and its rearranged form for finding an angle. Worked examples cover each step, including the obtuse-angle case, and the practice room gives unlimited randomised questions, all auto-marked to three significant figures.
What is the cosine rule?
The cosine rule is the other method for working in any triangle, not just right-angled ones. You reach for it in the two situations the sine rule cannot handle: when you know two sides and the angle between them, or when you know all three sides.
As with the sine rule, label each side with the lower-case letter of the angle opposite it: side \(a\) is opposite angle \(A\), and so on.
\[a^2 = b^2 + c^2 - 2bc\cos A\]
To find an ANGLE (you know all three sides):\[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]
Is this on the formula sheet? Yes. Both forms of the cosine rule, along with the sine rule and the area formula, are printed on the formula sheet at the front of the Edexcel IGCSE Maths A (4MA1) Higher tier paper, so you do not have to memorise them. What you do have to know is when to reach for the cosine rule and how to use it: choosing it over the sine rule, substituting correctly, and rearranging to find an angle. The marks come from applying it, not from recalling it, so the practice below focuses on exactly that.
When to use the cosine rule
The method
- Label the triangle so the angle you know (or want) is \(A\), and the side opposite it is \(a\).
- Choose the form: finding a side uses \(a^2 = b^2 + c^2 - 2bc\cos A\); finding an angle uses the rearranged \(\cos A\) form.
- Substitute carefully, keeping the negative sign in front of \(2bc\cos A\).
- Finish: for a side, take the square root; for an angle, use inverse cosine. Round to 3 significant figures only at the end.
Worked examples
💡 Example 1: finding a side (SAS)
In triangle ABC, b = 7 cm, c = 10 cm, and the included angle A = 120°. Find side a.
\[\begin{array}{rcl} a^2 &=& b^2 + c^2 - 2bc\cos A \\ &=& 7^2 + 10^2 - 2(7)(10)\cos 120^\circ \\ &=& 49 + 100 - 140(-0.5) \\ &=& 219 \end{array}\]
Taking the square root, \(a = 14.8\) cm (3 s.f.).
What's happening?
We know two sides and the angle between them, so this is the SAS case for the cosine rule.
The angle is obtuse, so \(\cos 120^\circ\) is negative. The two minus signs combine to add, which is why the answer comes out larger.
💡 Example 2: finding an angle (SSS)
In triangle ABC, a = 6 cm, b = 9 cm, and c = 11 cm. Find angle A.
\[\begin{array}{rcl} \cos A &=& \dfrac{b^2 + c^2 - a^2}{2bc} \\ &=& \dfrac{9^2 + 11^2 - 6^2}{2(9)(11)} \\ &=& \dfrac{166}{198} \end{array}\]
Taking the inverse cosine, \(A = \cos^{-1}(0.8383\ldots)\), so \(A = 33.0^\circ\) (3 s.f.).
What's happening?
All three sides are known, so this is the SSS case. We want angle \(A\), so the side opposite it, \(a\), goes in the \(-a^2\) position on top.
Once you have \(\cos A\), use inverse cosine to get the angle.
💡 Example 3: an acute included angle
In triangle ABC, b = 8 cm, c = 5 cm, and the included angle A = 52°. Find side a.
\[\begin{array}{rcl} a^2 &=& b^2 + c^2 - 2bc\cos A \\ &=& 8^2 + 5^2 - 2(8)(5)\cos 52^\circ \\ &=& 64 + 25 - 80(0.6157\ldots) \\ &=& 39.74\ldots \end{array}\]
Taking the square root, \(a = 6.30\) cm (3 s.f.).
What's happening?
This time the included angle is acute, so \(\cos 52^\circ\) is positive and the \(2bc\cos A\) term is subtracted as normal.
Keep the full calculator value through the working; only round the final length.
🔑 Key points
- Use the cosine rule for SAS (two sides and the included angle) or SSS (three sides).
- The side you find is always opposite the angle you use.
- Finding a side: \(a^2 = b^2 + c^2 - 2bc\cos A\).
- Finding an angle: \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\).
- Keep full accuracy and round only at the very end (3 s.f.).
⚠️ Common pitfalls
- Using the cosine rule when a side and its opposite angle are known (the sine rule is quicker there).
- Forgetting that \(\cos\) of an obtuse angle is negative, which flips the sign of the last term.
- Taking \(\sqrt{a^2}\) too early, or forgetting the square root altogether.
- Rounding partway through and losing accuracy.
- Calculator not in degree mode.