$t\mapsto (\cos t,\sin t)$ is a bijection from $[0,2\pi )$ onto the real unit circle

Depends on

• Parity and the Pythagorean identity for sine and cosine
• Signs, monotonicity intervals, and ranges of sine and cosine
• The zero sets of sine and cosine and the least positive common period 2 pi
• The subtraction formulas for sine and cosine
• Quarter-turn values and shifts by pi/2 and pi
• The derivatives of sine and cosine are cosine and minus sine
• Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$
• Pi as twice the smallest positive zero of cosine

Used by

• Buffered certified rational phase approximation Algorithm
• Every a cos x+b sin x has the amplitude-phase form R cos(x-φ) Corollary
• Spectral measures and the known-mean periodogram Definition
• The one-dimensional torus and its normalized Haar integral Definition
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• An atomic sinusoid and an AR(1) process have different spectra Example
• Annulus and punctured disc have hyperbolic universal covers Example
• Great circles as round-sphere geodesics Example
• Polar coordinates are a local diffeomorphism away from zero radius Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• SU(2) to SO(3) as a covering homomorphism Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
• The outward flux of the inverse-square field through a sphere centred at the origin is 4π Example
• A Quantitative Logarithm Near the Identity in SU(2) Lemma
• Box–Muller has a product Gaussian density and Rayleigh second moment Lemma
• Clifford+T Is Dense Projectively on One Qubit Lemma
• Determinant-one Clifford+T words are dense in SU(2) Lemma
• Determinant-one reduction for optimal Rz approximation Lemma
• Finite tori are compact Hausdorff spaces separated by characters Lemma
• Fourier uniqueness for continuous functions on the Euclidean torus Lemma
• Fresnel whole-line moment from the damped complex Gaussian Lemma
• Powers of an Irrational Rotation Are Dense on the Circle Lemma
• Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
• The exact operator-norm epsilon region for determinant-one Rz candidates Lemma
• The multiplicative unit circle is a compact metrizable topological abelian group Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• The projective one-qubit group is SO(3) Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• Two omitted finite values rule out an essential singularity Lemma
• Two-CNOT Synthesis up to a Migrating Terminal Diagonal Lemma
• Unoptimized recursive QSD decomposition and CNOT count Lemma
• [t]↦(cos 2π t,sin 2π t) is a homeomorphism from ℝ/ℤ to the unit circle Theorem
• A circle traversed k times has winding number k inside and 0 outside Theorem
• Correctness and arithmetic resource bound for buffered phase approximation Theorem
• Every nonzero complex number has a unique polar form r(cosθ+i sinθ) with r>0 and -π<θ≤π Theorem
• Fixed-Arity Unitary Error Budget Theorem
• ker(exp)=2π iℤ, and exp z=exp w exactly when z-w∈2π iℤ Theorem
• One-Qubit Euler Decomposition Theorem
• Phase estimation on a superposition of eigenspaces Theorem

…and 3 more results.

Dependency tree · two levels

37 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)