Deriving the Law of Cosines

Trigonometry · Axiom Academy

LESSON Deriving the Law of Cosines A coordinate proof that reveals how the Law of Cosines generalizes the Pythagorean theorem 1. Strategic Triangle Placement Consider a triangle with sides of length a , b , and c , where angle θ (theta) is opposite side c . Let's place this triangle on a coordinate system in a clever way: Put vertex C at the origin (0, 0) Place vertex A on the positive x-axis at (b, 0) Position vertex B at angle θ from the x-axis, distance a from the origin With our strategic placement, we can write the exact coordinates of each vertex using basic trigonometry: The coordinates of point B come from converting polar coordinates (distance a , angle θ) to Cartesian coordinates using the standard formulas. 3. Applying the Distance Formula Now we need to find the length of side c , which connects points A and B . We use the distance formula: Substituting our coordinates A = (b, 0) and B = (a cos θ, a sin θ) into the distance formula gives us a relationship we can simplify. Let's expand and simplify the distance formula step by step: After expanding the squared terms and using the Pythagorean identity (cos²θ + sin²θ = 1), we arrive at the Law of Cosines! The Law of Cosines truly is a generalization of the Pythagorean theorem. Watch what happens when the angle θ = 90°:

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