Pythagorean Identities

Calculus Readiness · Axiom Academy

One right triangle inside the unit circle generates an entire family of identities — and it s the Pythagorean theorem in disguise. 1. It s the Pythagorean Theorem Drop a point at angle on the unit circle. Its horizontal coordinate is and its vertical coordinate is , so it sits at the corner of a right triangle whose legs are and and whose hypotenuse is the radius 1 . Watch the squares grow on each side: the two leg-squares always fill exactly the square on the hypotenuse. a point at angle on the unit circle leg² + leg² = hypotenuse² (and the hypotenuse is 1) 2. Divide Through to Get the Other Two Every other identity comes from dividing the master equation by or . Watch the triangle dilate from the origin until the horizontal leg becomes exactly 1 : that s dividing every length by . The vertical leg becomes and the hypotenuse becomes , so the same theorem now reads . Each fits a different expression. See ? Reach for the tangent form. See ? Reach for the cotangent form. All three are the same Pythagorean theorem, just rescaled so a different side equals 1 . Each identity has two solved forms worth recognizing instantly: 3. Using It to Find Other Ratios The payoff: if you know one ratio and the quadrant, the identity hands you the rest. Suppose with in the third quadrant. Watch the horizontal leg grow in to close the right triangle back to hypotenuse 1 — its length is , and the quadrant fixes the sign. Step 1 — solve the master identity: .

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