A card from a free library — ask anything, no account, works offline. Every card carries its source.
From the unit circle to the double-angle law
mathematics
The Pythagorean identity is the unit circle written as algebra; the double-angle law for cosine follows in its train. Both are confirmed exactly (not sampled), for all x. Worked & sealed by the engine — sin²x + cos²x = 1 · cos 2x = 1 − 2 sin²x. [HOLDS; open the seal to re-check.]
source
card id
card_works_trig_identities
address
UNPLACED
adjoining cards
- on the shelf of → The Works — sealed, worked demonstrations — a member of the the-works shelf in the keeping
- kindred → Euler's identity, assembled from its parts — A sibling worked demonstration in mathematics.
- kindred → Differentiate twice: position → velocity → acceleration — A sibling worked demonstration in mathematics.
- paves → The Floor of Discovery — one floor, and by its design the fear of God — A worked, engine-sealed demonstration — a paving-stone of the floor of reality.
- demonstrates → cos(π) = −1 · sin(π) = 0 · e^{iπ} + 1 = 0 — A worked demonstration in the same field (mathematics).
- demonstrates → sin²x + cos²x = 1 · cos 2x = 1 − 2 sin²x — A worked demonstration in the same field (mathematics).
- demonstrates → d/dt (t³) = 3t² (position → velocity) · d/dt (3t²) = 6t (velocity → accelerati — A worked demonstration in the same field (mathematics).
- adjacent → Two laws of thought, checked over every case — Neighbouring worked demonstrations in the same part of the volume.
Is this card incomplete? Tell the library — it will call out for more ↗