De Moivre's Theorem

There is a beautiful theorem attached to our study of Complex numbers, and yes, it has a romantic name to it, like most other theorems in mathematics, known as De Moivre's Theorem. The theorem, though, is not as complicated as it sounds, but is definitely complex (pun intended). De Moivre's Theorem will serve as a gateway to more advanced theory of complex numbers and will help in tackling some crucial problems on the way. 

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How are the multiple-angle formulas for sine and cosine are derived from de moivre's theorem?

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What is a major application of de moivre's theorem in regard to solving equations?

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What is De Moivre's Theorem about?

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For a complex number \(z=r(\cos x+i \sin x)\), what will be \(z^n\) according to De moivre's theorem?

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Expand the complex number \(z^3=2^3(\cos \theta +i \sin\theta)^3\) without the use of binomial theorem.

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Does de moivre's theorem hold for negative powers of a complex number?

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For which of the following powers of a complex number, does the de moivre's theorem breaks down?

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Let \(x^n=a=ib\) where you can write \(a+ib=s(\cos \alpha +i \sin \alpha)\), then what will be the solution set to the equation \(x^n=a+ib\)?

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Which method is used to prove de moivre's theorem?

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If a complex is given of the form \(z=a+ib\), what are the steps to take the \(\text{nth}\) power of it?

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The principle argument of a complex number is \(\theta\). What will be the argument of a complex that is raised to an integer power \(k\)? 

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How are the multiple-angle formulas for sine and cosine are derived from de moivre's theorem?

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What is a major application of de moivre's theorem in regard to solving equations?

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What is De Moivre's Theorem about?

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For a complex number \(z=r(\cos x+i \sin x)\), what will be \(z^n\) according to De moivre's theorem?

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Expand the complex number \(z^3=2^3(\cos \theta +i \sin\theta)^3\) without the use of binomial theorem.

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Does de moivre's theorem hold for negative powers of a complex number?

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For which of the following powers of a complex number, does the de moivre's theorem breaks down?

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Let \(x^n=a=ib\) where you can write \(a+ib=s(\cos \alpha +i \sin \alpha)\), then what will be the solution set to the equation \(x^n=a+ib\)?

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Which method is used to prove de moivre's theorem?

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If a complex is given of the form \(z=a+ib\), what are the steps to take the \(\text{nth}\) power of it?

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The principle argument of a complex number is \(\theta\). What will be the argument of a complex that is raised to an integer power \(k\)? 

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    De Moivre's theorem definition

    Suppose you have a complex number \(z=1+2i\), and you are asked to calculate the square of this complex number: \(z^2=(1+2i)^2\).

    Well, easy enough:

    $$ \begin{aligned} z^2 & =(1+2i)(1+2i) \\ &=1+2i+2i-4 \\ z^2 &=-3+4i \end{aligned}$$

    But things get tedious as you go ahead, what if you were asked to calculate higher powers of it, like \(z^4, \ z^5,...\) and so on. Binomial expansion can be effectively used, but it becomes impractical as the powers get larger and larger.

    For such large powers, there is a very elegant theorem by which one can come up with a concise formula for any integer power of a complex number:

    De Moivre's Theorem (also known as De Moivre's identity) is named after the French mathematician Abraham de Moivre, although the theorem never appeared in any of his works.

    The statement of the theorem is given as follows:

    For a complex number \(z=r(\cos x +i\sin x)\), \(r\) being the modulus of the complex number and for some integer \(n\), raising \(z\) to the power of \(n\), we get:

    $$z^n=r^n(\cos {nx}+i \sin {nx})$$

    This is a mighty formula in itself since raising the power of a complex number to an integer, you simply have to multiply the argument of the complex by the integer the complex number is raised to.

    De Moivre's formula

    The formula for De Moivre's Theorem can be written down as follows:

    $$z^n=(r(\cos x+i \sin x))^n=r^n(\cos{nx}+i\sin{nx})$$

    To prove De Moivre's Formula, all you need is the method of 'Proof by Mathematical Induction':


    Proof:

    Let \(z=r(\cos x+i\sin x)\), and \(n \in \mathbb{N}\):

    Let \(P(n)\) be the statement that \(z^n=r^n(\cos{nx}+i\sin{nx})\).

    Proceed by proving \(P(1)\):

    $$ \begin{aligned} &\text{LHS}=z \\ & \text{RHS}=r(\cos x+i \sin x) \end{aligned}$$

    Thus, \(z=r(\cos x+i \sin x)\) (yes, it is as trivial as it seems), which renders \(P(1)\) to be true.

    Now assume that \(P(k)\) is true for \(\forall k \in \mathbb{N}\). So, you have \(z^k=r^k(\cos{kx}+i\sin{kx})\).

    The goal is to prove the statement \(P(k+1)\) using the fact that \(P(k)\) is true:

    $$z^{k+1}=r^{k+1}(\cos{(k+1)x}+i\sin{(k+1)x})$$

    Starting with \(\text{LHS}\):

    $$\begin{aligned} \text{LHS} &=z^{k+1} \\ &=z^k\ z \\ &=r^k(\cos x +i\sin x)^k\ r(\cos x+i\sin x) \end{aligned}$$

    Using the statement of \(P(k)\):

    $$z^k=r^k(\cos x +i\sin x)^k=r^k(\cos{kx}+i\sin{kx})$$

    Substituting this into \(\text{LHS}\):

    $$\begin{aligned} \text{LHS}&=r^{k+1}(\cos{kx}+i\sin{kx}) \ (\cos x+i\sin x) \\ &=r^{k+1}(\cos x\cos{kx} +i\cos{kx} \sin x +i\cos x \sin{kx} -\sin x \sin{kx}) \\ \therefore \ \text{LHS} &=r^{k+1}((\cos x \cos{kx} - \sin x \sin{kx})+i(\sin x \cos{kx} +\cos x \sin{kx}) ) \end{aligned}$$

    Now you just have to recognise the solution by recalling the trigonometric identities for the sine and cosine functions:

    $$\begin{aligned} &\cos{(a+b)}=\cos a \cos b -\sin a \sin b \\ &\sin{(a+b)}=\sin a \cos b+\cos a \sin b \end{aligned}$$

    Substituting these formulae into \(\text{LHS}\):

    $$\begin{aligned} \text{LHS} &=r^{k+1} (\cos{(k+1)x}+i\sin{(k+1)x}) \\ &=\text{RHS} \\ \therefore \ \text{LHS} &=\text{RHS}\end{aligned}$$

    which is what you were asked to prove.

    Hence, assuming \(P(k)\) to be true, it implies that \(P(k+1)\) is true. Thus, by the principle of mathematical induction, \(P(n)\) is true \(\forall n \in \mathbb{N}\).

    For all the negative integers, the theorem is yet to be proved. To do so, use the result you proved for the positive integers \(n\):

    $$z^n=r^n(\cos{nx}+i\sin{nx})$$

    Replacing \(n\) by \(-n\):

    $$\begin{aligned} z^{-n} &=r^{-n}(\cos (n x)+i \sin (n x))^{-1} \\ &=r^{-n}\frac{1}{\cos (nx)+i \sin (n x)} \cdot\left(\frac{\cos (n x)-i \sin( n x)}{\cos (n x)+i \sin (nx)}\right) \\ &=r^{-n}\frac{\cos(n x)-i \sin (n x)}{\cos^2 (nx)+\sin^2 (nx)} \\ &=r^{-n} \frac{\cos (-n x)+i(\sin (-n x))}{1} \\ \therefore \ z^{-n} & = r^{-n}(\cos(-n x)+i \sin (-n x)) \end{aligned}$$

    which proves the theorem for all the negative integers \(n\)

    Note that De Moivre's theorem does not hold for rational values of \(n\).

    Solving using De Moivre's theorem

    Suppose you are asked to solve the cubic equation \(x^3-1=0\), what would your answer be? It certainly would be \(x=1\), which is correct, but the answer is incomplete. \(x=1\) is the real root of this cubic equation, but what about the other roots? The degree of the polynomial is \(3\); hence it should have three roots in total.

    This where the methods you have learned in high school come to an end (kind of). It turns out that the other roots of this equation are not real, but complex.

    Let's try to address this generally and then go back to the problem of finding the cube root of \(1\). In general, we are trying to solve the equation \(x^n=a+ib\) where \(n \in \mathbb{N}\). Writing \(a+ib\) in polar form, we get:

    $$a+ib=s(\cos \alpha +i\sin \alpha) $$

    where \(s\) is the magnitude of \(a+ib\) and \(\alpha\) is the principal argument of it.

    Now, equating this to \(x^n\), but firstly, let \(x=r(\cos \theta +i\sin \theta)\), so that we have polar forms on both sides:

    $$\begin{aligned} &x^n = s(\cos \alpha +i\sin \alpha) \\ \therefore \ &(r(\cos \theta +i\sin \theta)^n = s(\cos \alpha +i\sin \alpha)\end{aligned}$$

    This where you use De Moivre's theorem on the left-hand side:

    $$ r^n(\cos {n\theta} +i \sin{n\theta}) = s(\cos \alpha +i\sin \alpha)$$

    Notice that \(s\) and \(\alpha\) are known, since \(a+ib\) is known. So \(r\) and \(\theta\) can be calculated by comparing the coefficients on both side as follows:

    $$ r=s^{1/n} \ \ \ \text{and} \ \ \ \theta=\frac{\alpha}{n}$$

    Now recall an important property of trigonometry, that sine and cosine functions are periodic i.e. \(\sin \theta =\sin{2\pi k+\theta}\) and \(\cos \theta = \cos{2 \pi k +\theta}\). Hence, in this case, \(\theta= \displaystyle \frac{\alpha +2 \pi k}{n}\).

    Therefore, you have \( r=s^{1/n}\) and \(\theta= \displaystyle \frac{\alpha +2 \pi k }{n}\), where \(k \in {1, \ 2, \ 3, ... , \ n-1}\).

    Let's get back to finding the cube roots of \(1\).

    Find the cube root of \(1\).

    Solution:

    Let \(x^3=1\) where you are asked to find the values of \(x\). One root can be observed straight away, which is \(x=1\).

    Writing \(1\) in polar form:

    $$1=1+0i=s(\cos \alpha +i\sin \alpha)$$

    Comparing the real and imaginary part, you have \(s=1\), \(\cos \alpha=1\) and \(\sin \alpha=0\) which gives \(\alpha=0\).

    Let the root be \(x=r(\cos \theta+i \sin \theta)\) so that,

    $$r=s^{1/n}=1 \ \ \text{and} \ \ \theta=\frac{\alpha +2 \pi k}{n}=\frac{2 \pi k}{3}$$

    Thus, you get the solution set as:

    $$x=\cos \left( {\frac{2 \pi k}{3}} \right) +i \sin \left( {\frac{2 \pi k}{3}} \right) $$

    All now left is to substitute \(k=0, \ 1, \ 2\) to have the different roots.

    For \(k=0\), \(x_1=1\), unsurprisingly, since it was the most obvious one.

    For \(k=1\),

    $$ \begin{aligned} x_2 & =\cos \left( \frac{2 \pi}{3} \right)+i\sin \left( \frac{2 \pi}{3} \right) \\ & = -\frac{1}{2}+i\frac{\sqrt{3}}{2} \\ \therefore \ x_2 & =\frac{-1+i\sqrt{3}}{2} \end{aligned}$$

    For \(k=2\),

    $$ \begin{aligned} x_3 & =\cos \left( \frac{4 \pi}{3} \right) + i \sin \left( \frac{4 \pi}{3} \right) \\ & = -\frac{1}{2}+i\frac{-\sqrt{3}}{2} \\ \therefore \ x_3 & =\frac{-1-i\sqrt{3}}{2} \end{aligned}$$

    Thus, the cube roots of \(1\) are \(1, \ \displaystyle \frac{-1+i\sqrt{3}}{2}, \ \frac{-1-i\sqrt{3}}{2}\).

    Application of De Moivre's theorem in expansion

    Try to think of some sort of application of De Moivre's theorem, just by looking at it. Let's pen the identity down for that:

    $$z^n=r^n(\cos x+i\sin x)^n=r^n(\cos{nx}+i\sin{nx})$$

    where \(z=r(\cos x+i\sin x)\). The identity essentially expands the complex number to give a simpler form, which is indeed the most important application of De Moivre's Theorem; expanding integer powers of a complex number.

    A complex number might be expressed in one of the two forms: \(z=a+ib\) and \(z=r(\cos x+i\sin x)\).

    The latter form is straightforward to work with; apply De Moivre's identity directly.

    Regarding the former form, convert it into the polar form, on which the identity can then be used. Let's take a look at some of the examples where this is in use.

    Find the value of \((\sin \left( \frac{\pi}{6} \right) + i \cos \left( \frac{\pi}{6} \right))^{18}\).

    Solution:

    Notice that the complex number is of the form \( \sin x+i \cos x\) and not \(\cos x+i \sin x\), so you need to convert it :

    $$\begin{aligned} \left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{18} &=\left(i\left(\cos \frac{\pi}{6}-i \sin \frac{\pi}{6}\right)\right)^{18} \\ &=i^{18}\left(\cos \frac{\pi}{6}-i \sin \frac{\pi}{6}\right)^{18} \end{aligned}$$

    Now you can safely apply de Moivre's identity:

    $$\begin{aligned} \left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{18} &=(-1)(\cos 3 \pi-i \sin 3 \pi) \\ &=-(-1-0 i) \\ &=1+0i \\ \therefore \ \left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{18} &=1 \end{aligned}$$

    Binomial theorem and De Moivre's theorem

    How would you write \(\sin 3\theta\) in terms of \(\sin \theta\) and \(\cos \theta\)? You shall explore a technique involving binomial theorem and de moivre's theorem that will answer this question. In general, you shall learn how to write \(\sin n\theta\) and \( \cos n\theta\) in terms of \(\sin \theta\) and \(\cos \theta\).

    Recall that the Binomial Theorem says that for a polynomial \((a+b)^n\) is expanded as:

    $$(a+b)^n=a^n + {n \choose 1} a^{n-1}b+ {n \choose 2} a^{n-2}b^2+...+b^n$$

    where \( \displaystyle {n \choose r} = \frac{n!}{r! \ (n-r)!} \).

    Using this, you can expand \((\cos \theta +i \sin \theta)^n\) to get:

    $$ \begin{aligned} (\cos \theta +i \sin \theta)^n &= \cos^n \theta+i{n \choose 1} \cos^{n-1} \theta \sin \theta+i^2{n \choose 2} \cos^{n-2} \theta \sin^2 \theta + \ ... \ +i^n \sin^n \theta \\ & = \left( \cos^n \theta -{n \choose 2} \cos^{n-2} \theta \sin^2 \theta +{n \choose 4} \cos^{n-4} \theta \sin^4 \theta + \ ... \right) \\ & \ \ \ \ +i\left( {n \choose 1} \cos^{n-1} \theta \sin \theta - {n \choose 3} \cos^{n-3} \theta \sin^3 \theta+ \ ...\right) \end{aligned} $$

    Comparing the real part of both sides:

    $$ \cos n\theta =\cos^n \theta -{n \choose 2} \cos^{n-2} \theta \sin^2 \theta +{n \choose 4} \cos^{n-4} \theta \sin^4 \theta + \ ...$$

    And now comparing the imaginary part:

    $$\sin n\theta = {n \choose 1} \cos^{n-1} \theta \sin \theta - {n \choose 3} \cos^{n-3} \theta \sin^3 \theta+ \ ...$$

    Thus, you have a formula with which the multiple-angle formulas for sine and cosine functions can be written down.

    De Moivre's theorem examples

    If \(z=(\cos \theta+i \sin \theta)\), show that \(z^n+\frac{1}{z^n}=2 \cos n \theta\) and

    \(z^n-\frac{1}{z^n}=2 i \sin n \theta\), where \(n \in \mathbb{Z}\)

    Solution:

    Let \(z=(\cos \theta+i \sin \theta)\), taking the \(n^{\text {th }}\) power:

    $$z^n=\cos n \theta+i \sin \theta$$

    Replacing \(n\) with \(-n\) :

    $$\begin{aligned} z^{-n} &=\cos(-n \theta)+i \sin (-n \theta) \\\therefore \ \frac{1}{z^n} &=\cos n \theta-i \sin n \theta \end{aligned}$$Adding \(z^n\) and \(\frac{1}{z^n}\):$$ \begin{aligned} z^n+\frac{1}{z^n} &=\cos n \theta+i \sin \theta+\cos n \theta-i \sin n \theta \\\therefore \ z^n+\frac{1}{z^n}&=2 \cos n \theta \end{aligned}$$

    Hence, the first identity has been proved.

    Subtracting \(1 / z^n\) from \(z^n\) :

    $$\begin{aligned} z^n-\frac{1}{z^n} &=\cos n \theta+i \sin n \theta-\cos n \theta+i \sin n \theta \\ \therefore \ z^n-\frac{1}{z^n}&=2 i \sin n \theta \end{aligned}$$

    Thus, the second identity has also been proved.

    Expand the complex number \((1-\sqrt{3} i)^5\) using De Moivre's identity.

    Solution:

    Since the complex number in not in polar form, we need to convert it first. To do so, we have to calculate the modulus of it, let \(z=1-i \sqrt{3}\) :

    $$\begin{aligned} r &=\sqrt{a^2+b^2} \\&=\sqrt{1+3} \\ \therefore \ r &=2 \end{aligned}$$

    For the argument of this complex number:

    $$\begin{aligned} \theta &=\tan ^{-1}\left(\frac{b}{a}\right) \\&=\tan ^{-1}\left(\frac{\sqrt{3}}{1}\right) \\\therefore \ \theta &=\pi / 3 \end{aligned}$$

    But be careful, this in not the argument. Notice that the complex number lies in the fourth quadrant, so:

    $$\begin{aligned}\therefore \phi &=2 \pi-\theta \\&=2 \pi-\frac{\pi}{3} \\\therefore \ \phi &=\frac{5 \pi}{3} \end{aligned}$$

    which is the correct argument of the complex number.

    Now, the polar form can be written down as:

    $$\begin{aligned} z &=r(\cos \phi+i \sin \phi) \\&=2\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right) \end{aligned}$$

    You now have the form on which you can apply De Moivre's identity:

    $$\begin{aligned} z^5 &=2^5\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right)^5 \\&=32\left(\cos \frac{25 \pi}{3}+i \sin \frac{25 \pi}{3}\right) \\\therefore \ z^5 &=32\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right) \\&=16+16 \sqrt{3} i \end{aligned}$$

    Thus, \((1-i \sqrt{3})^5=16+16 \sqrt{3} i\).

    De Moivre's Theorem – Key takeaways

    • De Moivre's Theorem states that a complex number \(z=r(\cos x+i\sin x)\), when raised to an integer power of \(n\), it can be written as \(z^n=r^n (\cos {nx} + i\sin {nx})\).
    • The theorem can be applied to finding roots of equations, for \(x^n=a+ib\), where \(a+ib)=s(\cos \alpha + i\sin \alpha)\). The solution of the equation \(x^n=a+ib\) is then given by \(x=s^{1/n} (\cos \left( \frac{\alpha + 2 \pi k}{n} \right) +i \sin \left( \frac{\alpha + 2 \pi k}{n} \right)) \).
    • The prime application of using De Moivre's Theorem is using it to expand a complex number of an integer power without the use of binomial theorem.
    Frequently Asked Questions about De Moivre's Theorem

    What is de moivre's theorem with example?

    De Moivre's theorem says that a complex number with an argument x, magnitude r, raised to a power of n will have magnitude r to the power of n, and argument nx. 

    An example is where z is raised to power 3 will have a new argument 3 times the original, with magnitude being raised to the power of 3.

    How does binomial theorem apply to de moivre's theorem?

    Binomial Theorem, in conjunction with De moivre's theorem, is used to get the multiple angle formulas for sine and cosine. 

    How to use de moivre's theorem to find roots?

     De moivre's theorem is mostly used to calculate the powers of a complex number, which can then be equated to another complex number to find the roots of it. 

    What is de moivre's theorem proof?

    De Moivre's theorem can be proved for positive integer powers by Proof of Mathematical Induction and then applying the result for negative integer powers. 

    How do you solve complex numbers using de Moivre's Theorem?

    De Moivre's is used to solve equations by finding the complex roots as well as expanding powers of a complex numbers without the need of binomial theorem. 

    Test your knowledge with multiple choice flashcards

    What is De Moivre's Theorem about?

    Does de moivre's theorem hold for negative powers of a complex number?

    For which of the following powers of a complex number, does the de moivre's theorem breaks down?

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