Search results
Results from the WOW.Com Content Network
A modest extension of the version of de Moivre's formula given in this article can be used to find the n-th roots of a complex number for a non-zero integer n. (This is equivalent to raising to a power of 1 / n). If z is a complex number, written in polar form as = ( + ),
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Chebyshev polynomials can be defined in this form when studying trigonometric polynomials. [4] That cos nx is an n th-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula: + = ( + ).
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [ 1 ] In the table below, the label "Undefined" represents a ratio 1 : 0. {\displaystyle 1:0.}
The values for a/b·2π can be found by applying de Moivre's identity for n = a to a b th root of unity, which is also a root of the polynomial x b - 1 in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts , respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i ...
de Moivre's theorem may be: de Moivre's formula, a trigonometric identity; Theorem of de Moivre–Laplace, a central limit theorem This page was last edited on 28 ...
De Moivre's theorem (complex analysis) De Rham's theorem (differential topology) Deduction theorem ; Dehn-Nielsen-Baer theorem (geometric topology) Denjoy theorem (dynamical systems) Denjoy–Carleman theorem (functional analysis) Denjoy-Young-Saks theorem (real analysis) Desargues's theorem (projective geometry) Descartes's theorem (plane ...