Principal inverse sine and inverse cosine

Depends on

• Signs, monotonicity intervals, and ranges of sine and cosine
• The derivatives of sine and cosine are cosine and minus sine
• A function differentiable at $c$ is continuous at $c$
• Quarter-turn values and shifts by pi/2 and pi
• Parity and the Pythagorean identity for sine and cosine
• Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$
• Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$

Used by

• Exact success amplification by attenuating the good branch Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• Szegedy-walk marked-set detection Algorithm
• Diameter rigidity from toponogov under a sectional lower bound Corollary
• A real-valued optimum is not a fractional Grover iteration Counterexample
• arcsin(sin x) is not the identity outside the principal interval Counterexample
• Principal arcsine has no finite derivative at -1 or 1 Counterexample
• Comparison angles of hinges, model triangle angles, and the Alexandrov upper angle Definition
• Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles Definition
• Spherical Gram simplices and angular links of Euclidean faces Definition
• The angular path metric, the Euclidean cone and spherical joins Definition
• The Coxeter nerve and its Moussong metric Definition
• The Logan-Shepp-Vershik-Kerov limit profile Ω Definition
• A circle of circumference ℓ<2π fails CAT(1) Example
• A complete locally CAT(0) circle whose fundamental group prevents global CAT(0) Example
• A spherical simplex from a Gram matrix and its vertex-link Schur complement Example
• Arcsine equilibrium measure and capacity of a segment Example
• Cut locus of a point on a round sphere Example
• Equally spaced points on a metric circle: stationary energy and the equality case Example
• Exact Grover rotation for four addresses and one marked item Example
• Link edge lengths versus dihedral mirror angles in type I₂(m) Example
• Midpoint iteration on a small equilateral spherical triangle contracts geometrically to its centre Example
• Null-homotopy versus shrinkability through short loops on S² and on a short circle Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers Example
• The unit circle is CAT(1) at the strict perimeter boundary Example
• Toponogov comparison on a round sphere Example
• A section curvature lower bound makes triangles thinner than the model False statement
• The cut locus of a point is always a smooth hypersurface False statement
• Alexandrov comparison: straightening a hinge, gluing comparison triangles, and patchwork Lemma
• Average Grover success for a uniform integer iteration count Lemma
• Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences Lemma
• Constant BBHT stage success beyond the threshold scale Lemma
• Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas Lemma
• Local CAT(1) of the l² product from a model S²× S² sine-comparison calculation Lemma
• Low-target amplification has a controlled multiplicative loss Lemma
• Minimum nonshrinkable loops, radial vertex cones, and the excursion of length π Lemma
• Orthogonal marked and unmarked components of the uniform state Lemma
• Products of CAT(0) spaces, joins of CAT(1) spaces, and round spheres Lemma
• The angular link of a vertex of the Davis complex is the large metric flag nerve Lemma
• The spherical radius estimate, the quadrilateral separation constant, and the finite midpoint-operation comparison disk Lemma

…and 16 more results.

Dependency tree · two levels

43 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, Inverse Trigonometric Functions (standard reference, not scraped)