Practice Law of cosines
The cosine rule gives c² = a² + b² − 2ab cos(γ), where γ is the angle between a and b.
It generalises the Pythagorean theorem to any triangle.
Exercises with solutions
The cosine rule states a² = b² + c² − 2 · b · c · cos α. It extends Pythagoras’ theorem: for α = 90°, cos α = 0 and the formula becomes a² = b² + c². You need it when two sides and the included angle, or all three sides, are known.
Exercise 1
In a triangle b = 5 cm, c = 7 cm and α = 60°. Calculate a.
Show solution and working
a ≈ 6.24 cm
a² = 25 + 49 − 2 · 5 · 7 · cos 60° = 74 − 70 · 0.5 = 39. So a = √39 ≈ 6.24 cm.
Exercise 2
What does the cosine rule give when the included angle is 90°?
Show solution and working
Pythagoras’ theorem
cos 90° = 0, so the term −2bc·cos α disappears and a² = b² + c² remains.
Exercise 3
In a triangle a = 6 cm, b = 5 cm and c = 7 cm. Calculate the angle γ.
Show solution and working
γ ≈ 78.5°
cos γ = (a² + b² − c²) ÷ (2ab) = (36 + 25 − 49) ÷ 60 = 12 ÷ 60 = 0.2. So γ ≈ 78.5°.
Exercise 4
When do you need the cosine rule instead of the sine rule?
Show solution and working
with two sides and the included angle, or with three sides
In those cases the pair of side and opposite angle that the sine rule requires is missing.
Common mistake
Using the cosine rule even though a pair of side and opposite angle is known — the sine rule is simpler then. And forgetting to use arccos when solving for the angle.
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