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27-708: The rabbit ( 兔 ) is the fourth in the twelve-year periodic sequence (cycle) of animals that appear in the Chinese zodiac related to the Chinese calendar . The Year of the Rabbit is associated with the Earthly Branch symbol 卯 . the element Wood in Wuxing theory and within Traditional Chinese medicine the Liver Yin and the emotions and virtues of kindness and hope . In
54-427: A cycle or orbit ) is a sequence for which the same terms are repeated over and over: The number p of repeated terms is called the period ( period ). A (purely) periodic sequence (with period p ), or a p- periodic sequence , is a sequence a 1 , a 2 , a 3 , ... satisfying for all values of n . If a sequence is regarded as a function whose domain is the set of natural numbers , then
81-422: A group . A periodic point for a function f : X → X is a point x whose orbit is a periodic sequence. Here, f n ( x ) {\displaystyle f^{n}(x)} means the n -fold composition of f applied to x . Periodic points are important in the theory of dynamical systems . Every function from a finite set to itself has a periodic point; cycle detection
108-420: A complex number raised to the integer power n . If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities ). 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
135-423: A periodic sequence is simply a special type of periodic function . The smallest p for which a periodic sequence is p -periodic is called its least period or exact period . Every constant function is 1-periodic. The sequence 1 , 2 , 1 , 2 , 1 , 2 … {\displaystyle 1,2,1,2,1,2\dots } is periodic with least period 2. The sequence of digits in
162-466: Is a rational number that equals p / q in lowest terms then this set will have exactly q distinct values rather than infinitely many. In particular, if w is an integer then the set will have exactly one value, as previously discussed.) In contrast, de Moivre's formula gives r w ( cos x w + i sin x w ) , {\displaystyle r^{w}(\cos xw+i\sin xw)\,,} which
189-472: Is a complex number, written in polar form as then the n -th roots of z are given by where k varies over the integer values from 0 to | n | − 1 . This formula is also sometimes known as de Moivre's formula. Generally, if z = r ( cos x + i sin x ) {\displaystyle z=r\left(\cos x+i\sin x\right)} (in polar form) and w are arbitrary complex numbers, then
216-424: Is clearly true. For our hypothesis, we assume S( k ) is true for some natural k . That is, we assume Now, considering S( k + 1) : See angle sum and difference identities . We deduce that S( k ) implies S( k + 1) . By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0 x ) + i sin(0 x ) = 1 + 0 i = 1 . Finally, for
243-401: Is easy to check the validity of the equation by multiplying out the left side. De Moivre's formula is a precursor to Euler's formula e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} with x expressed in radians rather than degrees , which establishes the fundamental relationship between
270-517: Is just the single value from this set corresponding to k = 0 . Since cosh x + sinh x = e , an analog to de Moivre's formula also applies to the hyperbolic trigonometry . For all integers n , If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x ) . For any integer n , the formula holds for any complex number z = x + i y {\displaystyle z=x+iy} where To find
297-404: Is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x . As written, the formula is not valid for non-integer powers n . However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the n th roots of unity , that is, complex numbers z such that z = 1 . Using
SECTION 10
#1732847610100324-403: Is the imaginary unit ( i = −1 ). The formula is named after Abraham de Moivre , although he never stated it in his works. The expression cos x + i sin x is sometimes abbreviated to cis x . The formula is important because it connects complex numbers and trigonometry . By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x
351-458: Is the algorithmic problem of finding such a point. Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions: One standard approach for proving these identities is to apply De Moivre's formula to the corresponding root of unity . Such sequences are foundational in
378-549: The Vietnamese zodiac and the Gurung zodiac , the cat takes the place of the rabbit . In the Malay zodiac , the mousedeer takes the place of the rabbit. People born within these date ranges can be said to have been born in the "Year of the Rabbit", while also bearing the following elemental sign: Periodic sequence In mathematics , a periodic sequence (sometimes called
405-400: The decimal expansion of 1/7 is periodic with period 6: More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below). The sequence of powers of −1 is periodic with period two: More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in
432-688: The cube roots are given by: With matrices, ( cos ϕ − sin ϕ sin ϕ cos ϕ ) n = ( cos n ϕ − sin n ϕ sin n ϕ cos n ϕ ) {\displaystyle {\begin{pmatrix}\cos \phi &-\sin \phi \\\sin \phi &\cos \phi \end{pmatrix}}^{n}={\begin{pmatrix}\cos n\phi &-\sin n\phi \\\sin n\phi &\cos n\phi \end{pmatrix}}} when n
459-433: The decimal expansion of 1/56 is eventually periodic: A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x 1 , x 2 , x 3 , ... is asymptotically periodic if there exists a periodic sequence a 1 , a 2 , a 3 , ... for which For example, the sequence is asymptotically periodic, since its terms approach those of
486-428: The identity of these parts can be written using binomial coefficients . This formula was given by 16th century French mathematician François Viète : In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact valid even for complex values of x , because both sides are entire (that is, holomorphic on
513-441: The negative integer cases, we consider an exponent of − n for natural n . The equation (*) is a result of the identity for z = cos nx + i sin nx . Hence, S( n ) holds for all integers n . For an equality of complex numbers , one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If x , and therefore also cos x and sin x , are real numbers , then
540-540: The periodic sequence 0, 1, 0, 1, 0, 1, .... De Moivre%27s formula In mathematics , de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity ) states that for any real number x and integer n it is the case that ( cos x + i sin x ) n = cos n x + i sin n x , {\displaystyle {\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,} where i
567-431: The right side is equal to cos n x + i sin n x . {\displaystyle \cos nx+i\sin nx.} The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n , call the following statement S( n ) : For n > 0 , we proceed by mathematical induction . S(1)
SECTION 20
#1732847610100594-408: The roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form can be represented in the form In this representation, and the trigonometric functions are defined as In the case that a + b + c ≠ 0 , that is, the unit vector. This leads to the variation of De Moivre's formula: To find the cube roots of write the quaternion in the form Then
621-591: The set of possible values is z w = r w ( cos x + i sin x ) w = { r w cos ( x w + 2 π k w ) + i r w sin ( x w + 2 π k w ) | k ∈ Z } . {\displaystyle z^{w}=r^{w}\left(\cos x+i\sin x\right)^{w}=\lbrace r^{w}\cos(xw+2\pi kw)+ir^{w}\sin(xw+2\pi kw)|k\in \mathbb {Z} \rbrace \,.} (Note that if w
648-1124: The standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is an arbitrary complex number. For x = 30 ∘ {\displaystyle x=30^{\circ }} and n = 2 {\displaystyle n=2} , de Moivre's formula asserts that ( cos ( 30 ∘ ) + i sin ( 30 ∘ ) ) 2 = cos ( 2 ⋅ 30 ∘ ) + i sin ( 2 ⋅ 30 ∘ ) , {\displaystyle \left(\cos(30^{\circ })+i\sin(30^{\circ })\right)^{2}=\cos(2\cdot 30^{\circ })+i\sin(2\cdot 30^{\circ }),} or equivalently that ( 3 2 + i 2 ) 2 = 1 2 + i 3 2 . {\displaystyle \left({\frac {\sqrt {3}}{2}}+{\frac {i}{2}}\right)^{2}={\frac {1}{2}}+{\frac {i{\sqrt {3}}}{2}}.} In this example, it
675-415: The study of number theory . A sequence is eventually periodic or ultimately periodic if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as a k + r = a k {\displaystyle a_{k+r}=a_{k}} for some r and sufficiently large k . For example, the sequence of digits in
702-401: The trigonometric functions and the complex exponential function. One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers since Euler's formula implies that the left side is equal to ( cos x + i sin x ) n {\displaystyle \left(\cos x+i\sin x\right)^{n}} while
729-527: The whole complex plane ) functions of x , and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for n = 2 and n = 3 : The right-hand side of the formula for cos nx is in fact the value T n (cos x ) of the Chebyshev polynomial T n at cos x . De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves
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