Introduction
The Law of Sines relates the sides and angles of any triangle (not just right triangles). It is particularly useful for solving oblique triangles where two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.
Formula
For a triangle with angles A, B, C and opposite sides a, b, c:
a / sin A = b / sin B = c / sin C
Strategy for Solving Triangles
- Identify which parts of the triangle are known (sides and angles).
- Choose the Law of Sines if you have AAS, ASA, or SSA.
- Set up the proportion using the known values.
- Use algebra to solve for the unknown side or angle.
- Check if there is an ambiguous case (SSA) that may produce two possible solutions.
Examples
Example 1 — Finding a Side:
Triangle ABC has A = 40°, B = 60°, and a = 8. Find side b.
Use Law of Sines: b / sin B = a / sin A → b / sin 60° = 8 / sin 40° → b ≈ 8 · sin 60° / sin 40° ≈ 10.97
Example 2 — Finding an Angle (SSA Case):
Triangle XYZ has x = 7, y = 10, and angle X = 30°. Find angle Y.
sin Y / y = sin X / x → sin Y / 10 = sin 30° / 7 → sin Y = 10 · 0.5 / 7 ≈ 0.714 → Y ≈ 45.57° Check for ambiguous case: Y′ = 180° − 45.57° ≈ 134.43° (possible if triangle angles sum < 180°)
Practice Problems
- Triangle PQR: P = 50°, Q = 70°, p = 12. Find side q.
- Triangle DEF: d = 9, e = 12, angle D = 40°. Find angle E and angle F.
- Triangle ABC: a = 15, c = 20, angle C = 60°. Find angle A and side b.
- Triangle LMN: L = 45°, M = 55°, side m = 10. Find side n.
- Triangle XYZ: x = 8, y = 6, angle X = 35°. Determine if there is an ambiguous case and solve.