The derivatives of sine and cosine are cosine and minus sine

Depends on

• Sine and cosine defined by their real power series
• The sine and cosine power series converge absolutely for every real argument
• Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius

Used by

• Exact success amplification by attenuating the good branch Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• Periodic Fourier collocation differentiation matrix Algorithm
• A uniform limit of smooth functions need not be differentiable anywhere Corollary
• Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin Corollary
• Diameter rigidity from toponogov under a sectional lower bound Corollary
• Every continuous function on [0,1] is uniformly approximated by everywhere-differentiable functions whose derivative vanishes at a prescribed point Corollary
• Parity and the Pythagorean identity for sine and cosine Corollary
• Sine and cosine are 1-Lipschitz on ℝ Corollary
• The limit of sin x divided by x at zero is one Corollary
• 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
• A curl-free C¹ field on the complement of a line that is not conservative Counterexample
• A figure-eight curve is an immersed image but not an embedded submanifold Counterexample
• A map with two preimages but degree zero Counterexample
• A real-valued optimum is not a fractional Grover iteration 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
• Backward heat amplifies small high-frequency errors Counterexample
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• Hadamard instability despite analytic solvability Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Irrational rotation is ergodic but not weakly mixing Counterexample
• Pointwise limit discontinuous at zero signals mass escape Counterexample
• Principal arcsine has no finite derivative at -1 or 1 Counterexample
• The circular curve defeats the equality form of the vector-valued mean value theorem Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• The vortex field is closed but not exact on the punctured plane Counterexample
• Uniform payoff convergence does not imply gradient convergence Counterexample
• Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable Counterexample
• Arcsine-weighted inner product on the Chebyshev interval Definition
• Finite-interval oscillatory integral and phase convention Definition
• Periodic Fourier collocation grid and even-grid Nyquist convention Definition
• Principal inverse sine and inverse cosine Definition
• Radian angle by unit-circle arc length Definition
• The Logan-Shepp-Vershik-Kerov limit profile Ω Definition
• The normalised sinc function Definition
• The one-dimensional torus and its normalized Haar integral Definition
• A closed cylinder as a finitely patched oriented surface Example

…and 208 more results.

Dependency tree · two levels

13 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)