De Moivre's theorem and its application





De-Moivre’s theorem







Let \( z=r \cos \theta +i\sin \theta \) be a complex number then \( z^n =r^n (\cos n\theta +i\sin n\theta )\) where n is a positive integer.
Proof

  1. Case 1: n=1
    Then
    \( z =r (\cos \theta +i\sin \theta )\)
    or \( z^1 =r^1 (\cos 1.\theta +i\sin 1.\theta )\)
    So,
    or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=1
  2. Case 2: n=2
    \( z^2 =z.z\)
    or \( z^2 = r (\cos \theta +i\sin \theta ) \times r (\cos \theta +i\sin \theta ) \)
    or \( z^2 = r^2 (\cos \theta +i\sin \theta )(\cos \theta +i\sin \theta )\)
    or \( z^2 = r^2 [\cos \theta (\cos \theta +i\sin \theta )+i\sin \theta (\cos \theta +i\sin \theta )\)
    or \( z^2 = r^2 [\cos \theta \cos \theta +i\cos \theta\sin \theta +i\sin \theta\cos \theta -\cos \theta\sin \theta ]\)
    or \( z^2 = r^2 [(\cos \theta \cos \theta-\sin \theta \sin \theta ) +i(\cos \theta\sin \theta +\sin \theta\cos \theta) ]\)
    or \( z^2 = r^2 [\cos (\theta +\theta) +i \sin (\theta+ \theta)]\)
    or \( z^2 = r^2 [\cos 2\theta +i \sin 2\theta]\)
    So,
    or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=2
  3. Case 3: We assume the same formula is true for n = k, so we have
    \( (\cos\theta + i\sin\theta)^k = r^k(\cos(k\theta) + i\sin(k\theta))\)
    So,
    or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=k
  4. Case 4: Now, we prove for n = k + 1,
    \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^k(\cos\theta + i\sin\theta)^k r (\cos\theta + i\sin\theta)\)
    or \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^k(\cos(k\theta) + i\sin(k\theta)) r(\cos\theta + i\sin\theta)\)
    or \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^{k+1}[(\cos(k\theta) \cos\theta-\sin(k\theta)\sin\theta )+i (\cos\theta\sin(k\theta) + \sin\theta\cos(k\theta))]\)
    or \( [r(\cos\theta + i\sin\theta)]^{k + 1} =r^{k+1}[\cos(k\theta+\theta)+i \sin(k\theta+\theta )]\)
    or \( [r(\cos\theta + i\sin\theta)]^{k + 1} =r^{k+1}[\cos(k+1)\theta+i \sin(k+1)\theta]\)
    So,
    or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=k+1
  5. Using case 1-case 4, for any number \( n \in Z\) , we have
    \( [r(\cos\theta + i\sin\theta)]^n =r^n[\cos(n\theta)+i \sin(n\theta)]\)



Exercise 1

  1. Evaluate
    1. \( [2 (\cos 15^o+i \sin 15^o)]^6\)

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    2. \( [3 (\cos 120^o+i \sin 120^o)]^3\)

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    3. \( [ (\cos 18^o+i \sin 18^o)]^5\)

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    4. \( [ (\cos 9^o+i \sin 9^o)]^{40}\)

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    5. \( (1+i)^6\)
      Solution

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    6. \( (1+i)^{20}\)

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    7. \( (-1+i)^{14}\)

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    8. \( \left ( \frac{1}{2} , \frac{\sqrt{3}}{2} i \right )^7 \)

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    9. \( (1-\sqrt{3}i)^6\)

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    10. \( i^2\)

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  2. Show that \(z^3=1\) where \( z=\left ( -\frac{1}{2} , \frac{\sqrt{3}}{2} i \right ) \)

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nth root of Complex number







If \( Z=r(\cos \theta +i\sin \theta )\) be a complex number then the nth root of z is
\( \sqrt[n]{Z}=\sqrt[n]{r} \left ( \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ) \)
Proof
Given that Z is a complex number. Also let, nth root of Z is W such that \( W=R(\cos \phi +i\sin \phi )\)
Now we have
\( \sqrt[n]{Z}=W\)
or \( W^n=Z\)
or \( [R(\cos \phi +i\sin \phi )]^n=r(\cos \theta +i\sin \theta )\)
or \( R^n(\cos (n\phi) +i\sin (n\phi) )=r(\cos \theta +i\sin \theta )\)
Equating real and Imaginary parts, we get
\( R^n=r\) and \( \cos (n\phi)= \cos \theta\) and \( \sin (n\phi) =\sin \theta \)
or \( R=\sqrt[n]{r}\) and \( n\phi= \theta +2k \pi \)
or \( R=\sqrt[n]{r}\) and \( \phi= \frac{(\theta +2k\pi )}{n}\)
Thus, nth root of \( Z=r(\cos \theta +i\sin \theta )\) is
\( W=R(\cos \phi +i\sin \phi )\)
or \( W=\sqrt[n]{r} \left ( \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ) \)




Exercise 2

  1. Find the square roots of
    1. \(i\)

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    2. \(4+4\sqrt{3} i\)

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    3. \(-1+\sqrt{3} i \)

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    4. \(-2-2\sqrt{3} i \)

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    5. \(2i\)

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    6. \(-i \)

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  2. Find the cube roots of
    1. \(8+6i\)

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    2. \(-1\)

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    3. \(8i\)

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  3. Solve the following
    1. \(z^4=1\)
    2. \(z^6=1\)
    3. \(z^4+1=0\)
    4. \(z^3=8i\)
  4. Find the 4th roots of \( ( -\frac{1}{2},\frac{\sqrt{3}}{2}i ) \)
  5. Find the 10th roots of 1
    The figure shows 10th root of 1.
    Drag the value of k=0,1,2,...,9
  6. In electrical engineering, a circuit has an impedance represented by the complex number. Z=8+6i ohms. The engineers need to design a component with an impedance that, when cubed, matches the original impedance.
    1. Calculate the magnitude ∣Z∣ and angle ΞΈ of the original impedance.
    2. Determine the cube root of the original impedance in polar form.
    3. Design a new component with an impedance Zn such that (Zn)^3 matches the original impedance.
    4. Express the new impedance in rectangular form and calculate its magnitude and angle.
  7. If \( \bar{z}\) be the conjugate of a complex number \(z\), prove that \(Arg(\bar{z})=2 \pi- Arg(z)\)
  8. If \(z=\cos theta +i \sin \theta\), prove that \(z^n-\frac{1}{z^n} = 2 \sin n \theta i\)



Properties of cube root of unity




Exponential Form

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