Two Sliding Rings: A Self-Checking Instrument that Reads Sine, Cosine and Tangent at One Setting

Two identical rings, each marked with the sixty repeating last digits of the Fibonacci numbers, are slid over each other until they cross on an angle. At that one setting the distance between the centres, halved, is the cosine and the distance between the crossings, halved, is the sine — the tangent is their ratio, and every reading is certified by the digits sitting under the crossings. Finer rings of the same digits reach any decimal angle, and run backward the instrument reads an angle from a length. A self‑checking way to read sine, cosine and tangent from geometry alone. Every value is verified by the script in the appendix.