Law of Sines and Law of Cosines

ACT Math · Axiom Academy

LESSON Law of Sines & Law of Cosines Solving non-right triangles on the ACT — the two laws that work when SOH-CAH-TOA can't. Label a triangle's angles A , B , C and put the side opposite each angle in lowercase: side a faces angle A, side b faces B, side c faces C. The Law of Sines says each side divided by the sine of its opposite angle gives the same value. Each side over the sine of its opposite angle — all equal. You can flip every fraction — use whichever form puts the unknown up top. AAS — two angles and a non-included side ASA — two angles and the included side SSA — two sides and an angle opposite one of them (the "ambiguous case") 2. Example 1 — Finding a Side (Law of Sines) Problem: In triangle ABC , angle , angle , and side a = 12 . Find side b . 3. Example 2 — Finding an Angle (Law of Sines) Problem: In triangle ABC , side a = 8 , side b = 11 , and angle . Find angle B . When you don't have a complete angle–side pair, the Law of Cosines takes over. It's a generalization of the Pythagorean theorem — watch: swing the included angle C and the closing side c changes. Right at the correction term vanishes ( ) and you're left with plain c^2 = a^2 + b^2 . The side you solve for is alone on the left; the cosine uses the angle opposite it. The formula works for any side — match each angle to the side across from it: SAS — two sides and the included angle → find the third side SSS — all three sides known → find any angle 5. Example 3 — Finding a Side (Law of Cosines)

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