Law of Sines

Pre-Calculus · Axiom Academy

In any triangle, each side divided by the sine of its opposite angle gives the same number — and that one fact solves triangles a right angle never could. 1. Every Side Has a Partner Across From It Label a triangle so side a is opposite angle A , side b opposite B , side c opposite C . The Law of Sines pairs each side with the angle it faces — and claims the ratio of a side to the sine of its partner angle is the same for all three pairs. side sine of its opposite angle — one shared value 2. Where It Comes From: One Altitude, Read Twice Drop a perpendicular of height h from vertex C straight down to side c . It splits the triangle into two right triangles that share that same height . Reading h off each one gives two expressions for the identical segment. Divide both sides by to free the sides: . An altitude from a different vertex repeats the argument for the remaining pair, so . No right angle in the original triangle was needed — we built the right angles ourselves. 3. Putting It to Work: Solve a Triangle Use the Law of Sines whenever you have a complete angle–side pair plus one more piece. Suppose , , and the side opposite A is a = 9 . That's AAS — two angles and a side — so the triangle is fully determined. Step 1 — find the third angle. The angles sum to , so . Step 2 — solve for the unknown sides. Anchor on the known pair , then multiply by the sine of each opposite angle. One full side–angle pair, plus a second angle (AAS / ASA) or a second side (SSA).

This is the written version of the interactive lesson above. See the full Pre-Calculus course.