Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions

Depends on

• Pythagorean and parity identities for all six trigonometric functions on their natural domains
• Quarter-turn values and shifts by pi/2 and pi
• The addition formulas for sine and cosine
• The zero sets of sine and cosine and the least positive common period 2 pi
• Tangent, cotangent, secant, and cosecant on their exact natural domains

Used by

• Buffered certified rational phase approximation Algorithm
• A fixed Grover iteration count can fail when a is unknown Counterexample
• arcsin(sin x) is not the identity outside the principal interval Counterexample
• I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
• A prepared state with success amplitude one half Example
• Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
• All meets and joins of the right weak order of A₂ (S₃), with the left order and the inversion sets compared Example
• B-tilde versus C-tilde: the n=2 coincidence and the duality behind the difference Example
• G2 from I2(6): the scaled realization and its twelve roots Example
• Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example
• Morrie's law: cos(π/9)cos(2π/9)cos(4π/9)=1/8 Example
• Quadratic Gauss sum at p=3 Example
• Quadratic Gauss sum at p=5 Example
• The A₂ root and weight lattices: P/Q has order three Example
• The arc length of one sine period is 4√2 E(1/√2) Example
• A Quantitative Logarithm Near the Identity in SU(2) Lemma
• Cartan-number products, allowed edge labels, tree scalings and reflection stability Lemma
• Continuous Markov derivative bound on an interval Lemma
• Efficient coherent amplitude-grid rounding Lemma
• Exact zero-label probability for counting Lemma
• Operator error for the finite sine-prefix rotation table Lemma
• Sine-window normalization and positive prefix weights Lemma
• The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices Lemma
• Two nearest phase labels carry probability at least eight over pi squared Lemma
• Amplitude-estimation additive error and success probability Theorem
• Classification of affine Coxeter diagrams and their Euclidean simplex realization Theorem
• Correctness and arithmetic resource bound for buffered phase approximation Theorem
• Known-amplitude success bound Theorem
• Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
• Success bound under an explicit amplitude-amplification phase mismatch Theorem
• Success probability and repetition for modular order finding Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)