Law of Cosines

Pre-Calculus · Axiom Academy

The Pythagorean theorem, generalized to every triangle — by one correction term. 1. Two Sides and the Angle Between Them The classic setup is SAS — you know two sides and the angle between them, and you want the third side. Pin side b down, swing side a up to open an angle C at the shared corner, and the gap that closes the triangle is the unknown side c . The Law of Cosines reads that gap straight off the two sides and the angle. The side you solve for sits alone on the left With a = 6 , b = 8 , and : , so . Two sides and one angle pin the whole triangle. 2. The Correction Term Does All the Work Look at the formula as Pythagoras plus a fix-up : c^2 = a^2 + b^2 would be the answer if C were a right angle, and corrects for the fact that it usually isn't. As the angle C opens wider, watch the third side c — and the correction term — respond. , so is negative — it pulls c^2 below a^2 + b^2 . The short third side of a squashed triangle. , the correction vanishes, and c^2 = a^2 + b^2 on the nose. , so is positive — it pushes c^2 above a^2 + b^2 . The long third side of a spread-open triangle. It is exactly the cross-term you would get squaring a and b as vectors — the cosine scales it by how much the two sides point the same way.

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