Cosine rule for sides and angles
Use an included angle to find a side, or use three sides to calculate an angle.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Label opposite triangle sides and angles.
- Use square roots and inverse cosine in degrees.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which angle is included between b and c?
What you will learn
- Apply a² = b² + c² − 2bc cos A.
- Rearrange the cosine rule to find an acute or obtuse angle.
Two sides and the angle between them
For sides b and c with included angle A, the opposite side satisfies a² = b² + c² − 2bc cos A. With b = 6, c = 8 and A = 60°, a² = 36 + 64 − 96 × 0.5 = 52. Therefore a = √52 = 2√13 cm, about 7.21 cm.
The included angle must lie between the two known sides. At A = 90°, cosine is zero and the rule becomes Pythagoras. For an obtuse included angle, cosine is negative, so subtracting its term increases a². Do not change the formula's minus sign to guess the result.
Three sides determine an angle
Rearrange to cos A = (b² + c² − a²)/(2bc). For a = 8, b = 5 and c = 7 cm, cos A = (25 + 49 − 64)/70 = 1/7. Inverse cosine gives A ≈ 81.8°. It returns the correct triangle angle on the interval zero to 180 degrees.
First check that the lengths can form a triangle: the sum of the two shorter sides must exceed the longest. Keep brackets around the entire numerator and denominator in a calculator. Compare the largest angle to the longest side and round a non-exact angle to one decimal place unless the question states otherwise.
Pause and explain
Sides 6 and 8 have included angle 60°. What is the opposite side squared?
Worked example
Two triangle sides are 6 and 8 cm with included angle 60°. Find the opposite side.
Show the worked solution
- a² = 6² + 8² − 2 × 6 × 8 cos 60° = 52.
- a = √52 = 2√13 cm.
- The positive root gives a ≈ 7.21 cm to three significant figures.
Answer 2√13 cm ≈ 7.21 cm.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Sides 3 and 4 cm have included angle 90°. Find the third side.
Give me a hint
Cosine ninety is zero.
Compare my reasoning
- a² = 9 + 16 − 24 × 0 = 25.
- a = 5 cm.
- This agrees with Pythagoras for a right triangle.
5 cm.
Look for these in your work
- Used the included angle.
- Recognised the right-angle special case.
A triangle has sides a = 8, b = 5 and c = 7 cm. Find A to one decimal place.
Give me a hint
Isolate cosine of the angle opposite side eight.
Compare my reasoning
- cos A = (25 + 49 − 64)/70 = 1/7.
- A = cos⁻¹(1/7) ≈ 81.7868°.
- Report 81.8°; side eight has the largest opposite angle.
81.8°.
Look for these in your work
- Chose the side opposite the requested angle.
- Bracketed the full quotient.
Sides 4 and 6 m meet at 120°. Find the opposite side exactly.
Give me a hint
Cosine one hundred twenty is negative one half.
Compare my reasoning
- a² = 16 + 36 − 48(−1/2) = 76.
- a = √76 = 2√19 m.
- Its square exceeds 4² + 6² because the included angle is obtuse.
2√19 m.
Look for these in your work
- Handled the negative cosine correctly.
- Checked the obtuse-angle interpretation.
Common mistakes
- Using a non-included angle.
- Missing brackets in the rearranged quotient.
- Ignoring invalid side lengths.
What can you explain now?
Can lengths 3, 4 and 8 cm form a triangle?
Compare with the explanation
No.
Three plus four is less than eight, so the triangle inequality fails before applying a rule.
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Make it stick
Tomorrow, decide between sine and cosine rules from the information given in two sketches.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.