Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

De-Moivre’s Theorem

Statement:

 (\cos \theta+i \sin \theta)^n=\cos n~ \theta+i \sin~ n ~\theta for any integer  n .

De-Moivre’s Theorem (for Rational Index)

Let  P \text { and } q be two integers with  q > 0,  \left( \cos \dfrac{P}{q}\theta+i \sin \dfrac{P}{q}\theta \right)^q=(\cos P ~\theta+i \sin~ P~ \theta)=(\cos \theta+i \sin \theta)^P

\begin{aligned}\to \left[ \cos \left( \dfrac{P ~\theta}{q}+\dfrac{2 \pi}{q} \right)+ i \sin\left( \dfrac{P~ \theta}{q}+\dfrac{2 \pi}{q} \right) \right]^q&=\cos (P ~\theta+2 \pi) +i \sin (P~ \theta+2\pi) \\&=\cos (P ~\theta)+ i \sin (P ~\theta)\\&=(\cos \theta +i \sin \theta)^P \end{aligned}

Example 1:

 \text { Value of }\begin{aligned}\left( \cos \dfrac{\pi}{5}+i \sin \dfrac{\pi}{5} \right)^5&=\cos \left( 5\times \dfrac{\pi}{5} \right)+i \sin \left( 5 \times \dfrac{\pi}{5} \right) \\&=\cos \pi+i \sin \pi=-1+i \times 0=-1 \end{aligned}

Example 2:

Solve:  \left( \cos \dfrac{3 \pi}{7}+i \sin \dfrac{3 \pi}{7} \right)^7

 =(\cos 3 \pi+i \sin 3 \pi)^{\frac{1}{7}\times7}

 =(\cos 3 \pi+i \sin 3 \pi)

 =(\cos \pi+i \sin \pi)^3

=(-1+i \times 0)^3=(-1)^3=-1

Principal Argument of  z in the interval  0 \leq \theta < 2 \pi

(i) In the interval 0 \leq \theta < \dfrac{\pi}{2}

(ii) In the interval  \dfrac{\pi}{2} < \theta \leq \pi

(iii) In the interval \pi \leq \theta < \dfrac{3\pi}{2}

(iv) In the interval  \dfrac{3 \pi}{2} < \theta \leq 2\pi

All of the angles ( \theta) in the interval  0\leq \theta < 2 \pi are in anti-clockwise direction.

Principal Argument of  z in the interval (-\pi < \theta \leq \pi)

(i) In the interval -\pi < \theta < -\dfrac{\pi}{2}

(ii) In the interval -\dfrac{\pi}{2}< \theta \leq \pi

All of the angles in the interval -\pi < \theta \leq \pi are in clockwise direction.

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