De Moivre's theorem
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Interpretation of De Moivre's Theorem and the n roots of unity
A graphical interpretation is as follows: Suppose that [Equation goes here - download the original to see it.]. For a definite illustrator let us consider [Equation goes here - download the original to see it.] . Then graphically we plot [Equation goes here - download the original to see it.] by 1The argument of [Equation goes here - download the original to see it.] is 3 times the argument of . 2 The modulus of [Equation goes here - download the original to see it.] is the cube of the modulus of [Diagram goes here - download the original to see it.] If r > 1 then the values of , , …. "spiral outwards": [Diagram goes here - download the original to see it.] If r < 1 then the values of , , …. "spiral inwards If r = 1 then the values of , , …. all lie on the unit circle. [Diagram goes here - download the original to see it.] The above illustration suggests that we can apply De Moivre's theorem in reverse to find solutions to the equation:- [Equation goes here - download the original to see it.] = 1 This is indeed the case. However, we observe that the equation [Equation goes here - download the original to see it.] = 1 has two solutions, x = i and x = -i Likewise, we expect the equation [Equation goes here - download the original to see it.] = 1 to have n solutions, and this is the case. In polar form the equation [Equation goes here - download the original to see it.] = 1 takes the form [Equation goes here - download the original to see it.] Applying De Moivre's theorem [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] One solution to the equation is = 0. However, we should recall that the angle 0 is given modulo [Equation goes here - download the original to see it.] and that [Equation goes here - download the original to see it.] Hence the n roots of unity - that is the n roots to the equation [Equation goes here - download the original to see it are given by the n distinct solutions to the equation [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The solutions in polar form are the n distinct complex numbers [1,0], [1 ], [1 ], …. For example, the solutions to [Equation goes here - download the original to see it.] are [1,0], [1 ], [1 ] Graphically, these are represented thus:- [Diagram goes here - download the original to see it.] In Cartesian form [Equation goes here - download the original to see it.] We can also use De Moivre's theorem to find solutions to equations such as [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] In Cartesian form: [Equation goes here - download the original to see it.]
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Contents of
De Moivre's theorem
1 De Moivre's Theorem
2 Interpretation of De Moivre's Theorem and the n roots of unity
3 Applications of De Moivre's Theorem to Trigonometric Identities
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