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Mathematics · Class 12 Science

Ch 2De Moivre's Theorem — Class 12 Mathematics, concept-first.

The previous chapter introduced the polar (or ) form of a complex number and showed that multiplying two complex numbers in this form is easy on the angles: . As an immediate consequence, squaring gives .

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De Moivre's Theorem

De Moivre's theorem states that (cosθ + isinθ)ⁿ = cos nθ + isin nθ for every integer n, and it extends (as a one-valued statement) to rational n as well. It is the single tool that lets us…

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Chapter contents

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Introduction

The previous chapter introduced the polar (or ) form of a complex number and showed that multiplying two complex numbers in this form is easy on the angles: .

2.1

De Moivre's Theorem for Integral and Rational Indices

We already know that multiplying two complex numbers written in the form is easy on the angles: , because and satisfy the angle-addition formulas. Squaring a single is just a special case of this: .

2.2

nth Roots of a Complex Number and nth Roots of Unity

A positive real number has exactly one positive real root, but complex numbers behave differently: a non-zero complex number has exactly distinct complex numbers satisfying , and all of them deserve t…

2.3

Cube Roots of Unity and Their Properties

The case of the previous section deserves its own name because it shows up so often in algebraic identities: the cube roots of unity.

2.4

Proving Trigonometric Identities Using (cos θ + i sin θ)^n

Besides finding roots, De Moivre's theorem is a genuine proof technique for trigonometric identities, especially ones involving multiple angles like or .

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 17 questions17 questions
  1. Q1If $(x - iy)^{1/3} = a - ib$, then show that: $\dfrac{x}{a} + \dfrac{y}{b} = 4(a^2 - b^2)$.Preview
  2. Q2If $n$ is a positive integer, show that: $(P + iQ)^{1/n} + (P - iQ)^{1/n} = 2(P^2 + Q^2)^{1/2n} \cdot \cos\left[\dfrac{1}{n}\tan^{-1}\dfrac{…Preview
  3. Q3If $(\sqrt{3} + i)^{100} = 2^{99}(a + ib)$, then show that $a^2 + b^2 = 4$.Preview
  4. Q4If $1, \omega, \omega^2$ are the cube roots of unity, then find the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$.Preview
  5. Q5If $\alpha, \beta$ are the roots of the equation $x^2 - 2x + 4 = 0$, then for any $n \in N$, show that $\alpha^n + \beta^n = 2^{n+1} \cdot \…Preview
  6. Q6If $\alpha, \beta$ are the roots of the equation $x^2 + x + 1 = 0$, then prove that $\alpha^4 + \beta^4 + \alpha^{-1}\beta^{-1} = 0$.Preview
  7. Q7If $\cos\alpha + \cos\beta + \cos\gamma = 0 = \sin\alpha + \sin\beta + \sin\gamma$, prove that : $$\cos^2\alpha + \cos^2\beta + \cos^2\gamma…Preview
  8. Q8If $1, \omega, \omega^2$ are the cube roots of unity, then find the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$.Preview
  9. Q9If $\alpha, \beta$ are the roots of the equation $x^2 - 2x + 4 = 0$, then for any $n \in N$ show that $\alpha^n + \beta^n = 2^{n+1}\cos\left…Preview
  10. Q10If $1, \omega, \omega^2$ are the cube roots of unity, then find the value of $(1-\omega+\omega^2)^5 + (1+\omega-\omega^2)^5$.Preview
  11. Q11Show that one value of $\left[\dfrac{1+\sin\frac{\pi}{8} + i\cos\frac{\pi}{8}}{1+\sin\frac{\pi}{8} - i\cos\frac{\pi}{8}}\right]^{8/3}$ is $-…Preview
  12. Q12If A, B, C are angles of a triangle such that $x=\text{cis}A$, $y=\text{cis}B$, $z=\text{cis}C$, then find the value of $xyz$.Preview
  13. Q13If $n$ is an integer then show that $(1+i)^{2n}+(1-i)^{2n}=2^{n+1}\cos\frac{n\pi}{2}$.Preview
  14. Q14If $(\sqrt{3} + i)^{100} = 2^{99}(a + ib)$, show that $a^2 + b^2 = 4$.Preview
  15. Q15Find the cube roots of $8$.Preview
  16. Q16If $n$ is an integer then show that $(1 + i)^{2n} + (1 - i)^{2n} = 2^{n+1} \cos \dfrac{n\pi}{2}$.Preview
  17. Q17Show that one value of $\left[\dfrac{1 + \sin\frac{\pi}{8} + i\cos\frac{\pi}{8}}{1 + \sin\frac{\pi}{8} - i\cos\frac{\pi}{8}}\right]^{8/3}$ i…Preview