Introduction
The Law of Cosines is a generalization of the Pythagorean theorem for any triangle. It is particularly useful for solving triangles when you know two sides and the included angle (SAS) or all three sides (SSS).
Formulas
For triangle ABC with sides a, b, c opposite angles A, B, C:
- a² = b² + c² − 2bc cos A
- b² = a² + c² − 2ac cos B
- c² = a² + b² − 2ab cos C
Strategy for Solving Triangles
- Identify known sides and angles (SAS or SSS).
- Choose the correct formula based on the side you want to find.
- Substitute known values and solve for the unknown side or angle.
- If solving for an angle, use the inverse cosine function.
- Check your solution to ensure triangle angles sum to 180°.
Examples
Example 1 — Finding a Side (SAS):
Triangle ABC has b = 8, c = 6, and angle A = 60°. Find side a.
a² = b² + c² − 2bc cos A → a² = 8² + 6² − 2·8·6·cos 60° → a² = 64 + 36 − 96·0.5 → a² = 64 + 36 − 48 → a² = 52 → a ≈ 7.21
Example 2 — Finding an Angle (SSS):
Triangle XYZ has sides x = 7, y = 10, z = 12. Find angle X.
x² = y² + z² − 2yz cos X → 49 = 100 + 144 − 2·10·12·cos X → 49 = 244 − 240 cos X → 240 cos X = 195 → cos X ≈ 0.8125 → X ≈ 35.6°
Practice Problems
- Triangle PQR: P = 40°, q = 9, r = 12. Find side p.
- Triangle DEF: sides d = 7, e = 10, f = 12. Find angle D.
- Triangle ABC: a = 5, b = 8, C = 120°. Find side c.
- Triangle LMN: sides l = 6, m = 9, n = 11. Find all angles.
- Triangle XYZ: sides x = 10, y = 6, z = 8. Find angle Z.