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(cos x + i sin x)n = cos (nx) + i sin (nx). |
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:(cos x + i sin x)n = cos (nx) + i sin (nx). |
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The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. |
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The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. |
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useful expressions for cos(nx) and sin(nx) in terms of sin(x) and cos(x). |
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useful expressions for cos(nx) and sin(nx) in terms of sin(x) and cos(x). Furthermore, one can use the formula to find explicit expressions for the n-th roots of unity: complex numbers z such that zn = 1. |
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De Moivre's formula was first proved by Roger Cotes; it can be derived from (but historically preceded) Euler's Theorem eix = cos x + i sin x |
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De Moivre was a good friend of Newton; in 1698 he wrote that the formula had been known to Newton as early as 1676. It can be derived from (but historically preceded) Euler's formula eix = cos x + i sin x |
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(eix)n = einx. |
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(eix)n = einx (see exponential function). |
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The complex number eix = cos x + i sin x is oftentimes refered to as cis x for short. |
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De Moivre's formula is actually true more generally than stated above: if z and w are complex numbers, then (cos z + i sin z)w is a [multivalued function]? while cos (wz) + i sin (wz) is not, and one can state that |
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De Moivre's formula is actually true for all complex numbers x and all real numbers n, but this requires careful extension of several functions to the complex plane. <-- Loisel I'm not so sure this makes any sense. |
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:cos (wz) + i sin (wz) is one value of (cos z + i sin z)w. /Talk |
The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry.
By expanding the left hand side and then comparing real and imaginary parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of sin(x) and cos(x). Furthermore, one can use the formula to find explicit expressions for the n-th roots of unity: complex numbers z such that zn = 1.
De Moivre was a good friend of Newton; in 1698 he wrote that the formula had been known to Newton as early as 1676. It can be derived from (but historically preceded) Euler's formula eix = cos x + i sin x and the exponential law (eix)n = einx (see exponential function).
De Moivre's formula is actually true more generally than stated above: if z and w are complex numbers, then (cos z + i sin z)w is a [multivalued function]? while cos (wz) + i sin (wz) is not, and one can state that
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