Geometry · GCSE Maths
The cosine rule
GCSE Maths cosine rule: use a² = b² + c² − 2bc cos A to find a side from two sides and the included angle, or rearrange for an angle when all three sides are known.
Two sides and the angle between them, or all three sides: cosine rule. a² = b² + c² − 2bc cos A. If A is obtuse, cos A is negative, so −2bc cos A becomes an extra plus.
The important bits
What you need to know
- 1
The cosine rule: a² = b² + c² − 2bc cos A, where A is the angle between sides b and c, and a is the side opposite A.
- 2
Use it for SAS (two sides and the included angle) to find the third side, and for SSS (three sides) to find an angle.
- 3
To find an angle, rearrange: cos A = (b² + c² − a²) / (2bc), then A = cos⁻¹(…). This is the form you need for SSS.
- 4
If A is acute, cos A is positive, so a² is less than b² + c² (compare Pythagoras). If A is obtuse, cos A is negative, so a² is greater than b² + c².
- 5
When A = 90°, cos 90° = 0 and the cosine rule becomes Pythagoras. That is a useful check, not a reason to use cosine rule on a right-angled triangle first.
- 6
Label a as the side you are finding, or as the side opposite the angle you are finding. Mixing the letters so that 2bc uses the wrong pair is the usual algebraic disaster.
- 7
Keep the square until you have a², then square-root at the end for a length. For an angle, do not square-root; you want cos A, then inverse cosine.
- 8
After one angle from SSS, the other angles can use the sine rule, which is shorter, provided you watch the ambiguous case if the remaining angle might be obtuse.
Quotations worth analysing
Short evidence. Real method.
“a² = b² + c² − 2bc cos A”
A is the included angle between b and c. Writing −2ab cos C with the wrong pair is a labelling error, not a different version of the formula.
“cos A = (b² + c² − a²) / (2bc)”
Use this when all three sides are known. The minus sits with a², the side opposite the angle you want. Swapping a² onto the other side of the minus finds a different angle.
“Two sides and the included angle”
Included means the angle is between those two sides. Two sides and a non-included angle is sine rule (and possibly ambiguous), not cosine rule first.
Go deeper
SAS: find the opposite side
Sides 7 cm and 10 cm enclose an angle of 40°. The unknown side a is opposite 40°. a² = 7² + 10² − 2 × 7 × 10 × cos 40° = 49 + 100 − 140 cos 40°. cos 40° ≈ 0.7660, so 140 × 0.7660 ≈ 107.24, and a² = 149 − 107.24 = 41.76, a = √41.76 ≈ 6.46 cm. Side a should be the shortest because it sits opposite the smallest given angle, and 6.46 < 7 < 10 fits. If the included angle had been 120°, cos 120° = −0.5, so −2bc cos A = −140 × (−0.5) = +70, and a² = 149 + 70. The opposite side is larger, as an obtuse angle demands.
Go deeper
SSS: find an angle, watching the minus
Sides a = 8, b = 5, c = 6. To find angle A (opposite 8): cos A = (5² + 6² − 8²) / (2 × 5 × 6) = (25 + 36 − 64) / 60 = (−3) / 60 = −0.05. A = cos⁻¹(−0.05) ≈ 92.9°. The negative cosine tells you immediately that A is obtuse. Students who compute 64 − 25 − 36 in the numerator have swapped the minus and will get an acute angle opposite the longest side, which is geometrically wrong. Longest side opposite largest angle is the size check: 8 is the longest side, so A must be the largest angle, and here it is just over 90°. After A, use the sine rule for B, but remember B is acute, so the calculator’s sin⁻¹ is fine. Finding all three angles with three cosine-rule calculations works but is slow and accumulates rounding.
Go deeper
Choose cosine rule or sine rule in ten seconds
Count the data. Three sides: cosine rule for one angle. Two sides and the angle between them: cosine rule for the third side. Two angles and a side: sine rule (and 180° for the third angle). Two sides and a non-included angle: sine rule, with the ambiguous-case warning. Right angle plus two pieces: SOHCAHTOA or Pythagoras. Write the chosen formula with letters from your diagram before the numbers go in. If a question is in two parts, part (a) often unlocks the included angle or the opposite pair that part (b) needs. Area ½bc sin A uses the same included-angle picture as the cosine rule’s SAS case; if you have just found A from SSS, you can then find the area without dropping a perpendicular.
See the idea in action
In triangle ABC, b = 7 cm, c = 10 cm, angle A = 40°. Find side a to 3 s.f. Step 1: Two sides and the included angle A, so cosine rule. a is opposite A. Step 2: a² = b² + c² − 2bc cos A = 7² + 10² − 2 × 7 × 10 × cos 40°. Step 3: 49 + 100 − 140 cos 40° = 149 − 140 × 0.766044… = 149 − 107.246… = 41.754… Step 4: a = √41.754… = 6.4617… = 6.46 cm to 3 s.f. Check: 40° is acute and smaller than the other angles are likely to be, so a should be the shortest side: 6.46 < 7 < 10.
Exam technique
Turn knowledge into marks
Write a² = b² + c² − 2bc cos A with your numbers substituted before you press any keys. For an angle, isolate cos A first; do not square-root the cosine-rule formula.
Common mistakes
Do not give these marks away
- 01
Using 2ab or 2ac in the last term when the included angle is A, so the pair should be 2bc.
- 02
Putting the wrong side in the minus when rearranging for cos A, so the largest angle comes out acute.
- 03
Taking the square root too early, or using the sine rule first on SAS data that has no opposite pair yet.
Which data set needs the cosine rule first?
ATwo sides and the included angle
BTwo angles and any side
CA right angle, one side and one other acute angle
DTwo sides and the angle opposite one of them, both angles acute and a clearly unique triangle
Show the answer
Two sides and the included angle. SAS is cosine rule for the third side. Two angles and a side is sine rule. A right angle with a side and an acute angle is SOHCAHTOA. Two sides and a non-included angle is sine rule (SSA).
Quick questions
If this is the bit you searched
When do I use the cosine rule?
When you know two sides and the included angle (to find the third side), or when you know all three sides (to find an angle).
How do I find an angle with the cosine rule?
cos A = (b² + c² − a²) / (2bc), then A = cos⁻¹ of that value. Side a is the one opposite the angle you want.
What happens if the angle is obtuse?
cos A is negative, so −2bc cos A is positive and a² is larger than b² + c². Inverse cosine of a negative number correctly returns an obtuse angle.
Is the cosine rule allowed in a right-angled triangle?
Yes: cos 90° = 0 and it becomes Pythagoras. Use Pythagoras or SOHCAHTOA first when you can see the right angle; they are shorter.