TrigoSystems

Modules · Module 05

Laws of sines and cosines

Once a triangle is no longer right, the primary ratios are not enough. The two laws extend the same ideas to any triangular system.

a / sin A = b / sin B = c / sin C c² = a² + b² − 2ab cos C

Law of sines

a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius.

Use when you have AAS, ASA, or the SSA ambiguous case.

Ambiguous case (SSA)

Given two sides and a non-included acute angle, there may be no triangle, one triangle, or two.

Compare the given side opposite the angle with the height h = other side × sin(angle). That comparison decides the count.

Law of cosines

c² = a² + b² − 2ab cos C, and cyclic permutations.

Use for SAS (find the third side) or SSS (find an angle). When C = 90°, it collapses to Pythagoras.