Choosing the Right Law

Pre-Calculus · Axiom Academy

One question decides it: do you have an angle paired with its opposite side? 1. The One Question: Is There a Complete Pair? Label a triangle the standard way: side a sits opposite angle A , side b opposite B , side c opposite C . The Law of Sines links each angle to the side across from it: Law of Sines — needs a known angle and its opposite side To use it you need at least one complete pair : an angle together with the side opposite it. The animation takes one fixed triangle and walks through all five ways you might be handed three parts. Watch for the moment a bright link snaps between an angle and the side across from it — that link is a usable pair, and it means Law of Sines. 2. No Pair? Then Law of Cosines Two patterns refuse to give you a complete pair. In SAS the known angle sits between the two known sides, so it is opposite the one side you don't know. In SSS you have three sides and zero angles. Neither lets Law of Sines start — and that is exactly when Law of Cosines takes over. Law of Cosines — links all three sides to one angle Use Law of Cosines to find the third side first. Now you have a side and its opposite angle, so you can finish with Law of Sines. Rearrange Law of Cosines to solve for an angle directly: . It needs an angle across from a known side. In SAS the angle is across from the unknown side; in SSS there's no angle at all. When , and the law collapses to c^2=a^2+b^2 . The term is the correction for a non-right angle.

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