Using Law of Cosines (SSS)

Pre-Calculus · Axiom Academy

EXAMPLE Using Law of Cosines (SSS) Finding all three angles of a triangle when only its three sides are known A triangle has sides a = 5 , b = 7 , and c = 9 . Find all three angles A , B , and C , where each angle is opposite the side of the same letter. Nice work. You solved a full SSS triangle: three sides in, three angles out — no ambiguity. Largest angle first: the angle opposite the longest side is the biggest, so finding it first via avoids any sign ambiguity later. A negative cosine means obtuse: here , so is obtuse — only one angle in a triangle can be. Finish with the angle sum: once two angles are known, the third is just , no extra Law-of-Cosines work needed. Result: , , , and they sum to . A valid SSS triangle always satisfies the triangle inequality ( ); the Law of Cosines then pins down every angle exactly.

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