Pythagorean Identity

Algebra 2 · Axiom Academy

LESSON The Pythagorean Identity The one identity behind every other — and it's just the Pythagorean theorem, hiding on the unit circle. 1. The Pythagorean Theorem in Disguise Drop a point at angle on the unit circle. Its coordinates are , so the right triangle beneath it has a horizontal leg of length , a vertical leg of length , and a hypotenuse equal to the radius, 1 . The Pythagorean theorem does the rest. The point lies on the circle of radius 1… …which is exactly the Pythagorean identity. The identity is not just for acute angles in a triangle — it holds for every , all the way around the circle. As the point orbits, and constantly trade size, but their sum never budges from 1 . 3. Using It: Recover the Missing Function Because , knowing one of or pins down the other up to a sign — and the quadrant supplies the sign. Given with in Quadrant II, find . In Quadrant II the x -coordinate is negative, so we take the negative root: . 4. Two More Identities for Free Divide the whole identity through by , then again by , and two more identities drop out — the ones you reach for with , , , and . The Pythagorean identity is just the Pythagorean theorem living on the unit circle — and everything else in this lesson followed from it. Scroll up to revisit any step.

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