10. Trigonometry & Complex Numbers

De Moivre's theorem

De Moivre's theorem bridges complex analysis and trigonometry by stating that $(\cos \theta + i \sin \theta)^n = \cos(n \theta) + i \sin(n \theta)$ for any integer $n$.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

De Moivre's Theorem for Rational Indices

For any real number $\theta$ and for any integer $n$, $$(\cos \theta + i \sin \theta)^n = \cos(n \theta) + i \sin(n \theta)$$. Furthermore, if $n$ is a rational number $p/q$ where $q \in \mathbb{N}$ and $\gcd(p, q) = 1$, then $\cos(n \theta) + i \sin(n \theta)$ is one of the values of $(\cos \theta + i \sin \theta)^n$.

📐 Geometric Interpretation

In the Argand plane, multiplication of complex numbers corresponds to addition of arguments and multiplication of moduli. Raising a complex number of unit modulus (i.e., on the unit circle) to the power $n$ rotates its position vector by an angle $n \theta$ relative to the positive real axis.

⚙️ Algebraic & Structural Meaning

It establishes a fundamental isomorphism between the cyclic group of rotations under multiplication and the additive group of angles modulo $2\pi$. It converts powers of trigonometric binomials into linear combinations of multiple angles.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Real Argument $\theta$	`N/A`	Strictly Mandatory / Necessary	If $\theta$ is complex, the theorem in its standard form fails unless generalized via Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$.
Integer or Rational Exponent $n$	`N/A`	Strictly Mandatory	If $n$ is irrational or complex, $(\cos \theta + i \sin \theta)^n$ becomes a multi-valued function, and direct application without proper branch cuts leads to paradoxes.
Principal Branch & Periodicity	`N/A`	Essential for root extraction	Omitting the $2k\pi$ periodicity when extracting roots via $n$-th roots of unity results in missing $n-1$ distinct roots.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Sum of $n$-th Roots of Unity

$$1 + \alpha + \alpha^2 + \dots + \alpha^{n-1} = 0$$ where $\alpha = e^{i 2\pi / n}$

Rule: The sum of all $n$-th roots of unity is always zero for $n \ge 2$.

Application: Frequently asked in UP PGT to evaluate sums like $1 + \omega + \omega^2$ or higher order cyclic sums.

Product of $n$-th Roots of Unity

$$1 \cdot \alpha \cdot \alpha^2 \dots \alpha^{n-1} = (-1)^{n-1}$$

Rule: The product of all distinct $n$-th roots of unity evaluates directly to $(-1)^{n-1}$.

Reciprocal Sum Simplification

$$(x + \frac{1}{x})^n = 2\cos(n\theta) \text{ when } x = \cos\theta + i\sin\theta$$

Rule: Instantly convert binomial powers of trigonometric ratios into multiple-angle sums.

Application: Standard pattern in UP TGT for evaluating integrals or expansions of $\cos^n \theta$.

Sum of Angles in Roots

$$\sum_{k=0}^{n-1} \cos\left(\theta + \frac{2k\pi}{n}\right) = 0$$

Rule: Symmetrically distributed angles on a circle sum to zero for both sine and cosine components.

Application: Appears in advanced trigonometric series summation in UP PGT.

Modulus-Argument Form Conversion

$$(1 + \cos\theta + i\sin\theta)^n = \left(2\cos\frac{\theta}{2}\right)^n \left(\cos\frac{n\theta}{2} + i\sin\frac{n\theta}{2}\right)$$

Rule: Factor out half-angles immediately to simplify complex binomial expressions.

Rapid Elimination via Degree Checking

$$ \text{Degree of resulting polynomial} = n $$

Rule: Eliminate options where trigonometric multiple angles do not match the index $n$.

Application: Used for rejecting incorrect options instantly in multiple-choice questions.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Ignoring Multi-Valuedness for Rational Indices

Common Pitfall: Treating $(\cos\theta + i\sin\theta)^{p/q}$ as having only a single value instead of $q$ distinct values.

Counterexample: $$(1)^{1/3} = 1 \text{ (ignoring } e^{i 2\pi/3} \text{ and } e^{i 4\pi/3})$$

Why it fails: Failing to include $2k\pi$ ($k = 0, 1, \dots, q-1$) causes candidates to lose marks when asked for 'all values'.

🛡️ Exam Defense: Always append $+ 2k\pi$ to the argument before applying fractional indices.

⚠️ COMMON ERROR #02

Misapplication with Minus Signs

Common Pitfall: Applying De Moivre's theorem directly to $(\cos\theta - i\sin\theta)^n$ without adjusting the sign.

Counterexample: $(\cos\theta - i\sin\theta)^n = [\cos(-\theta) + i\sin(-\theta)]^n = \cos(n\theta) - i\sin(n\theta)$

Why it fails: Directly writing $\cos(n\theta) + i\sin(n\theta)$ for negative sine terms violates the standard form.

🛡️ Exam Defense: Ensure the expression is strictly in the form $(\cos\theta + i\sin\theta)$ with a plus sign before applying the theorem.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
De Moivre's Theorem (Integer)	$n \in \mathbb{Z}$, $\theta \in \mathbb{R}$	$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$	Rotation and scaling on unit circle	Powers of complex numbers or trigonometric expansions
De Moivre's Theorem (Rational)	$n = p/q$, $\gcd(p,q)=1$	$q$ distinct values via $2k\pi$	Vertices of a regular polygon inscribed in a circle	Find 'all values of' or 'cube roots / $n$-th roots'
Euler's Formula	$\theta \in \mathbb{R}$	$e^{i\theta} = \cos\theta + i\sin\theta$	Mapping of real line onto unit circle in complex plane	Exponential simplification and differential equations
Roots of Unity	$z^n - 1 = 0$, $n \in \mathbb{N}$	$z_k = e^{i 2k\pi / n}$ for $k=0,1,\dots,n-1$	Vertices of regular $n$-gon with vertex at $1$	Sum or product of roots questions

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $x = \cos\theta + i\sin\theta$, find the value of $x^n + \frac{1}{x^n}$.

(A) $2\cos(n\theta)$(B) $2\sin(n\theta)$(C) $\cos(n\theta) + i\sin(n\theta)$(D) $2i\sin(n\theta)$

📚 Method A: Formal Textbook Derivation

Step 1: Write $x = \cos\theta + i\sin\theta$. Step 2: By De Moivre's theorem, $x^n = \cos(n\theta) + i\sin(n\theta)$ and $\frac{1}{x^n} = \cos(n\theta) - i\sin(n\theta)$. Step 3: Add both expressions to get $2\cos(n\theta)$ térm.

⚡ Method B: 25-Second Exam Shortcut

Direct standard identity recognition. Memorize $x + 1/x = 2\cos\theta$ and $x - 1/x = 2i\sin\theta$.

💡 Tip: Direct formula recall item frequently featured in UP TGT.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the sum of the roots of the equation $z^5 - 1 = 0$?

📚 Method A: Formal Textbook Derivation

Step 1: The roots are the 5th roots of unity: $1, \alpha, \alpha^2, \alpha^3, \alpha^4$. Step 2: By the theory of equations, the sum of the roots of $z^n - 1 = 0$ is the coefficient of $z^{n-1}$, which is $0$.

⚡ Method B: 25-Second Exam Shortcut

Sum of all $n$-th roots of unity for $n \ge 2$ is always zero.

💡 Tip: Do not compute individual roots; use coefficient comparison or symmetry.

Practice questions

Choose an answer to check it and see the solution:

If $x_r = \cos\left(\frac{\pi}{2^r}\right) + i \sin\left(\frac{\pi}{2^r}\right)$, then the value of $x_1 x_2 x_3 \dots \infty$ is:

A $-1$

B $1$

C $i$

D $-i$

Derivation & Analysis:

The product of arguments is a geometric series sum: $\frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + \dots = \pi$. Thus the product evaluates to $\cos\pi + i\sin\pi = -1$.

The value of $(1 + i\sqrt{3})^3 + (1 - i\sqrt{3})^3$ is:

A $8$

B $-8$

C $16$

D $-16$

Derivation & Analysis:

Convert to polar form: $1 + i\sqrt{3} = 2(\cos(\pi/3) + i\sin(\pi/3))$. Cubing gives $8(\cos\pi + i\sin\pi) = -8$. Similarly, $(1 - i\sqrt{3})^3 = -8$. Their sum is $-16$.

If $\alpha$ is an imaginary cube root of unity, then $(1 - \alpha + \alpha^2)^3 + (1 + \alpha - \alpha^2)^3$ equals:

A $8$

B $-8$

C $16$

D $-16$

Derivation & Analysis:

Using $1 + \alpha + \alpha^2 = 0$, we have $1 + \alpha^2 = -\alpha$, so $1 - \alpha + \alpha^2 = -2\alpha$. Cubing gives $(-2\alpha)^3 = -8\alpha^3 = -8$. Similarly, $(1 + \alpha - \alpha^2)^3 = -8$. Sum = $-16$ wait, let's check: $(-2\alpha)^3 = -8$ and $(-2\alpha^2)^3 = -8$ gives $-16$. Correct option index is 2 ($16$ or $-16$, let's verify signs: $(-2\alpha)^3 + (-2\alpha^2)^3 = -8 - 8 = -16$. Thus index 3).

The number of distinct values of $(\cos\theta + i\sin\theta)^{3/4}$ is:

A $3$

B $4$

C $12$

D Infinitely many

Derivation & Analysis:

For a fractional index $p/q$ in lowest terms, the number of distinct values is given by the denominator $q$, which is $4$.

If $z = \cos\theta + i\sin\theta$, then $z^n - \frac{1}{z^n}$ is equal to:

A $2\cos(n\theta)$

B $2i\sin(n\theta)$

C $\cos(n\theta) - i\sin(n\theta)$

D $i\sin(n\theta)$

Derivation & Analysis:

By De Moivre's theorem, $z^n = \cos(n\theta) + i\sin(n\theta)$ and $1/z^n = \cos(n\theta) - i\sin\theta$. Subtracting them yields $2i\sin(n\theta)$.

The roots of the equation $(z - 1)^3 = 8$ are:

A $3, -1 \pm i\sqrt{3}$

B $1, 1 \pm 2i$

C $3, 1 \pm i\sqrt{3}$

D $-3, 1 \pm i\sqrt{3}$

Derivation & Analysis:

Let $z-1 = y$, then $y^3 = 8 \implies y = 2(1)^{1/3}$. The cube roots of unity are $1, \omega, \omega^2$. Thus $y = 2, 2\omega, 2\omega^2$. Adding $1$ gives $z = 3, 1+2(-1/2 + i\sqrt{3}/2) = 1 - 1 + i\sqrt{3} = 1 + i\sqrt{3}$ and $1 - i\sqrt{3}$.

If $\omega$ is a complex cube root of unity, then the value of $(1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8)$ is:

A $0$

B $1$

C $-1$

D $3$

Derivation & Analysis:

Simplify terms using $\omega^3 = 1$: $\omega^4 = \omega$ and $\omega^8 = \omega^2$. The product becomes $(1+\omega)(1+\omega^2)(1+\omega)(1+\omega^2) = (-\omega^2)(-\omega)(-\omega^2)(-\omega) = \omega^3 \cdot \omega^3 = 1 \cdot 1 = 1$.

If $\alpha, \beta, \gamma$ are the roots of $z^3 - a = 0$, then $\alpha \beta \gamma$ is equal to:

A $a$

B $-a$

C $0$

D $1$

Derivation & Analysis:

By the theory of equations for $z^3 - a = 0$, the product of the roots is $(-1)^3 ( -a ) = a$ (or constant term with proper sign).

The real part of $(\cos\theta + i\sin\theta)^n$ is:

A $\sin(n\theta)$

B $\cos(n\theta)$

C $-\cos(n\theta)$

D $-\sin(n\theta)$

Derivation & Analysis:

By De Moivre's theorem, $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$. The real part is $\cos(n\theta)$.

Q10

If $x + \frac{1}{x} = 2\cos\theta$, then the value of $x^3 + \frac{1}{x^3}$ is:

A $2\cos(3\theta)$

B $2\sin(3\theta)$

C $\cos(3\theta)$

D $6\cos\theta$

Derivation & Analysis:

Setting $x = \cos\theta + i\sin\theta$, we get $x^3 + 1/x^3 = 2\cos(3\theta)$ directly from De Moivre's expansion.

Common questions

Directly applying the standard formula when $n$ is complex leads to multi-valued ambiguities. It must be evaluated using the exponential definition $z^n = e^{n \log z}$.

Failing to account for all $n$ distinct roots when dealing with fractional indices like $1/n$, or ignoring the $2k\pi$ periodicity term.

Memorize standard results such as the sum and product of roots of unity, expansions for $(x \pm 1/x)^n$, and recognize polar forms instantly without full manual expansion.

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Applications of De Moivre's theorem