The zero sets of sine and cosine and the least positive common period 2 pi

Depends on

• Pi is the first positive zero of sine
• Quarter-turn values and shifts by pi/2 and pi
• Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$
• The principle of mathematical induction
• Integer powers $a^m$
• Laws of integer exponents

Used by

• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• 1/sin(1/z) has a nonisolated singularity at 0 Counterexample
• A map with two preimages but degree zero Counterexample
• A surjective map need not be a fibration Counterexample
• A twice-traversed circle has the same trace but twice the path length Counterexample
• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• Polar coordinates on a full closed angular period are not injective and are singular at radius zero Counterexample
• sin(1/x) has no limit as x tends to zero Counterexample
• The circular curve defeats the equality form of the vector-valued mean value theorem Counterexample
• The topologist's sine curve is connected but not path connected Counterexample
• Tangent, cotangent, secant, and cosecant on their exact natural domains Definition
• The one-dimensional torus and its normalized Haar integral Definition
• ∑ₙ₌₁^∞sin(nx)/n converges pointwise but not uniformly Example
• A positive non-log-convex solution of the Gamma functional equation Example
• F(x)=x² sin(1/x²) has an unbounded derivative whose Henstock–Kurzweil integral is sin 1 Example
• Hopf circle fibration Example
• Mobius band as an interval bundle with monodromy Example
• Morrie's law: cos(π/9)cos(2π/9)cos(4π/9)=1/8 Example
• Polar coordinates are a local diffeomorphism away from zero radius Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam Example
• The four Dini derivatives of x sin(1/x) at 0 take two distinct values Example
• The Fourier series of a sawtooth and the Basel sum Example
• The Fourier series of a square wave and the odd reciprocal-square sum Example
• The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
• The solid generated by rotating y=sin x on [0,π] has volume π²/2 Example
• x² sin(1/x²) has an unbounded, non-Riemann-integrable derivative Example
• xy sin(1/(x²+y²)) is differentiable at the origin with unbounded partial derivatives nearby Example
• FALSE: an invertible derivative at one point gives a local inverse False statement
• False: any positive zero of sine characterizes pi False statement
• FALSE: spherical coordinates are globally injective False statement
• Finite sums of the sine harmonics Lemma
• Finite tori are compact Hausdorff spaces separated by characters Lemma
• Fourier uniqueness for continuous functions on the Euclidean torus Lemma
• Nearest-integer probe points for the Weierstrass function Lemma
• Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• The topologist's sine curve is connected Lemma
• The trigonometric characters are orthonormal in L² of the torus Lemma

…and 9 more results.

Dependency tree · two levels

34 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)
• C. Schmeiser, Introduction to Analysis (standard reference, not scraped)