Using Law of Cosines (SAS)

Pre-Calculus · Axiom Academy

EXAMPLE Using Law of Cosines (SAS) Given two sides and the angle between them, find the third side — then the rest of the triangle. In triangle ABC , side a = 7 , side b = 10 , and the included angle . Find side c . In triangle DEF , side d = 15 , side e = 20 , and angle . Find side f and the remaining angles D and E . Two ships leave a port at the same time. Ship A travels 30 miles on a bearing of N E, while Ship B travels 40 miles on a bearing of N E. How far apart are the two ships? — the side opposite the known angle, in terms of the other two sides and that angle. Nice work. You used the Law of Cosines to solve three SAS triangles — from a bare side, to a full triangle, to a navigation problem. SAS → third side: when you know two sides and the angle between them, gives the side opposite that angle. Next angle, use cosine again: rearranged, . Solving for the angle opposite the shortest side keeps it acute, so there is no ambiguous case. Last angle is free: the three angles sum to , so . Result: ; ; the ships are about 43.35 miles apart. Whenever a problem hands you two sides and the angle wedged between them, reach for the Law of Cosines first — it turns SAS into a fully solved triangle.

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