10. Trigonometry & Complex Numbers

Complex numbers

Complex numbers extend the real number system by introducing the imaginary unit $i$, satisfying $i^2 = -1$, forming an algebraically closed field $\mathbb{C}$ equipped with a rich geometric interpretation in the Argand plane. This topic covers modulus, argument, De Moivre's theorem, and root extraction.

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 rational number $n$, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$. When $n$ is a rational number $\frac{p}{q}$ (where $p, q \in \mathbb{Z}, q > 0$), $\cos(n\theta) + i \sin(n\theta)$ is one of the values of $(\cos \theta + i \sin \theta)^{p/q}$.

📐 Geometric Interpretation

Multiplication of two complex numbers in polar form scales their moduli and adds their arguments; raising a complex number to the $n$-th power scales its radius by $r^n$ and rotates its vector in the Argand plane by an angle $n\theta$.

⚙️ Algebraic & Structural Meaning

Bridges trigonometric identities with polynomial algebra, enabling efficient computation of powers and roots of unity.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Polar Representation Validity	`$\theta \in [-\pi, \pi]$ or $[0, 2\pi)$`	Strictly Mandatory	Failure to use principal arguments leads to multi-valued ambiguity and incorrect root extraction.
Modulus Non-Zero ($r > 0$)	$z \neq 0$	Strictly Mandatory for Polar/Exponential forms	Argument $\arg(0)$ is undefined, collapsing polar conversions.
Rationality of Index $n$	$n \in \mathbb{Q}$	Essential for finite root sets	Irrational exponents yield infinitely many values, requiring complex branch analysis.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Sum of $n$-th Roots of Unity

$$ \sum_{k=0}^{n-1} e^{i 2\pi k/n} = 0 \quad \text{and} \quad \prod_{k=0}^{n-1} (1 - \omega^k) = n $$

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

Application: Directly tested in UP PGT and KVS PGT for evaluating polynomial roots and symmetric sums.

Modulus Property of Complex Fractions

$$ \left| \frac{az + b}{\bar{b}z + \bar{a}} \right| = 1 \quad \text{when } |z| = 1 $$

Rule: If $|z| = 1$, bilinear transformations of this specific conjugate-symmetric form maintain unit modulus.

Triangle Inequality Extremes

$$ |z_1| - |z_2| \le |z_1 \pm z_2| \le |z_1| + |z_2| $$

Rule: Maximum distance occurs when vectors are collinear and codirectional; minimum occurs when opposite.

Application: Used for finding maximum and minimum values of moduli combinations in competitive tests.

Rotation of Complex Numbers

$$ z_2 - z_0 = (z_1 - z_0) e^{i \theta} $$

Rule: To rotate a point $z_1$ about center $z_0$ by angle $\theta$ counterclockwise, multiply the shifted vector by $e^{i\theta}$.

Application: Standard geometry problems involving equilateral triangles or squares on the Argand plane.

Real and Imaginary Power Extraction

$$ (1 \pm i)^n = 2^{n/2} \left( \cos\left(\frac{n\pi}{4}\right) \pm i \sin\left(\frac{n\pi}{4}\right) \right) $$

Rule: Convert binomials like $1+i$ or $\sqrt{3}+i$ instantly to polar form before applying powers.

Locus Identification Trick

$$ |z - z_1| = |z - z_2| \implies \text{Perpendicular bisector of segment } z_1 z_2 $$

Rule: Recognize standard distance forms to instantly mark straight lines, circles, or ellipses without algebraic expansion.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Principal Argument Miscalculation

Common Pitfall: Using $\tan^{-1}(y/x)$ blindly without checking the quadrant in which the complex point $(x, y)$ lies.

Counterexample: $z = -1 - i$

Why it fails: $\tan^{-1}((-1)/(-1)) = \tan^{-1}(1) = \pi/4$, but $z$ is in the third quadrant, so $\arg(z) = -3\pi/4$.

🛡️ Exam Defense: Always sketch the point in the Argand plane before determining its argument.

⚠️ COMMON ERROR #02

Power Law Distribution on Non-Real Bases

Common Pitfall: Assuming $(z_1 z_2)^{1/n} = z_1^{1/n} z_2^{1/n}$ without accounting for all branch values.

Counterexample: Roots of $(ab)^{1/n}$ versus $a^{1/n} b^{1/n}$

Why it fails: Radicals are set-valued functions in complex analysis; naive splitting misses $n-1$ other roots.

🛡️ Exam Defense: Always use exponential polar form $r e^{i(\theta + 2k\pi)}$ when extracting roots.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Cartesian vs Polar Form	$z = x + iy = r(\cos\theta + i\sin\theta)$	$r = \sqrt{x^2+y^2}, \theta = \text{atan2}(y,x)$	Vector magnitude and directional angle	Conversion questions
De Moivre's Theorem	$n \in \mathbb{Q}, \theta \in \mathbb{R}$	$(\cos\theta+i\sin\theta)^n = \cos n\theta + i \sin n\theta$	Scaling radius and rotating angle	Powers of complex numbers
Roots of Unity	$z^n - 1 = 0$	$z_k = e^{i 2\pi k/n}, k=0,1,\dots,n-1$	Vertices of a regular $n$-gon inscribed in unit circle	Sum or product of root problems
Triangle Inequality	$z_1, z_2 \in \mathbb{C}$	$\|\|z_1\| - \|z_2\|\| \le \|z_1+z_2\| \le \|z_1\| + \|z_2\|$	Triangle side length constraints	Max/min modulus problems

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $\omega$ is an imaginary cube root of unity, then the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ is:

(A) $32$(B) $-32$(C) $64$(D) $-64$

📚 Method A: Formal Textbook Derivation

Step 1: Use $1 + \omega + \omega^2 = 0$. Step 2: Rewrite $1 + \omega^2 = -\omega$ and $1 + \omega = -%\omega^2$. Step 3: Substitute into expression to get $( -2\omega )^5 + ( -2\omega^2 )^5$. Step 4: Simplify using $\omega^3 = 1$ to obtain $-32(\omega^2 + \omega) = -32(-1) = 32$ (Wait, let's recalculate: $(-2\omega)^5 = -32\omega^2$ and $(-2\omega^2)^5 = -32\omega^{10} = -32\omega$. Sum is $-32(\omega^2 + \omega) = -32(-1) = 32$).

⚡ Method B: 25-Second Exam Shortcut

Replace binomial parts instantly using $\omega$ properties, factor out $-32$, and use $\omega+\omega^2=-1$.

💡 Tip: Keep core identities $1+\omega+\omega^2=0$ and $\omega^3=1$ memorized.

CANONICAL PROBLEM 02 Dual-Method Breakdown

The locus of complex number $z$ satisfying $\left| \frac{z - i}{z + i} \right| = 1$ in the complex plane is:

(A) A circle(B) The real axis(C) The imaginary axis(D) A straight line parallel to real axis

📚 Method A: Formal Textbook Derivation

Step 1: Set $|z - i| = |z + i|$. Step 2: Let $z = x + iy$. Step 3: $|x + i(y-1)| = |x + i(y+1)|$. Step 4: Square both sides: $x^2 + (y-1)^2 = x^2 + (y+1)^2 \implies -2y = 2y \implies y = 0$. Step 5: $y=0$ represents the real axis.

⚡ Method B: 25-Second Exam Shortcut

The equation $|z - z_1| = |z - z_2|$ represents the perpendicular bisector of the line joining $z_1 = i$ and $z_2 = -i$. The perpendicular bisector of the imaginary axis segment between $i$ and $-i$ is the real axis ($y=0$).

💡 Tip: Instantly recognize geometric loci without expanding Cartesian coordinates.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $z = x + iy$ and $|z + 1| = |z - 1|$, then the locus of $z$ is:

A Real axis

B Imaginary axis

C Circle of radius 1

D Parabola

Derivation & Analysis:

$|x+1 + iy| = |x-1 + iy| \implies (x+1)^2 + y^2 = (x-1)^2 + y^2 \implies 4x = 0 \implies x = 0$, which is the imaginary axis.

MCQ 2: The amplitude of $\frac{1 + i \sqrt{3}}{1 + i}$ is:

A $\frac{\pi}{12}$

B $\frac{\pi}{6}$

C $\frac{\pi}{3}$

D $\frac{5\pi}{12}$

Derivation & Analysis:

$\arg(1+i\sqrt{3}) = \pi/3$, $\arg(1+i) = \pi/4$. Argument of quotient is difference: $\pi/3 - \pi/4 = \pi/12$.

MCQ 3: If $\omega$ is a non-real cube root of unity, then $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$ equals:

A $1$

B $2$

C $4$

D $8$

Derivation & Analysis:

Since $1+\omega^2 = -\omega$, first factor is $(-2\omega)$. Since $1+\omega = -\omega^2$, second factor is $(-2\omega^2)$. Product is $4\omega^3 = 4$.

MCQ 4: The value of $\sum_{k=1}^{10} (i^k + i^{k+1})$ where $i = \sqrt{-1}$ is:

A $0$

B $1+i$

C $-1+i$

D $i$

Derivation & Analysis:

Sum is a geometric series of powers of $i$. Simplifying cycles of 4 yields $-1+i$.

MCQ 5: If $|z| = 4$ and $\arg(z) = \frac{5\pi}{6}$, then $z$ in cartesian form is:

A $-2\sqrt{3} + 2i$

B $2\sqrt{3} - 2i$

C $-2 + 2\sqrt{3}i$

D $2 + 2\sqrt{3}i$

Derivation & Analysis:

$z = 4(\cos(5\pi/6) + i \sin(5\pi/6)) = 4(-\frac{\sqrt{3}}{2} + i\frac{1}{2}) = -2\sqrt{3} + 2i$.

MCQ 6: The number of solutions to the equation $z^2 + \bar{z} = 0$ in $\mathbb{C}$ is:

A $1$

B $2$

C $3$

D $4$

Derivation & Analysis:

Let $z = x+iy$. Substituting $z^2 + \bar{z} = 0$ and separating real and imaginary parts gives 4 distinct solutions: $0, 1, -1/2 + i\sqrt{3}/2, -1/2 - i\sqrt{3}/2$.

MCQ 7: If $z = \frac{1}{1 - \cos\theta + i \sin\theta}$, then $\text{Re}(z)$ is equal to:

A $\frac{1}{2}$

B $\frac{1}{2}\cot(\theta/2)$

C $\frac{1}{2}\tan(\theta/2)$

D $1$

Derivation & Analysis:

Rationalize denominator or use half-angle identities to find $\text{Re}(z) = 1/2$.

MCQ 8: If $1, \omega, \omega^2$ are the cube roots of unity, then $(1 - \omega)(1 - \omega^2)(1 - \omega^4)(1 - \omega^5)$ equals:

A $3$

B $9$

C $27$

D $81$

Derivation & Analysis:

Since $\omega^3 = 1$, $\omega^4 = \omega$ and $\omega^5 = \omega^2$. The expression evaluates to $(1-\omega)^2(1-\omega^2)^2 = [ (1-\omega)(1-\omega^2) ]^2 = (3)^2 = 9$.

MCQ 9: The minimum value of $|z| + |z - 3i|$ for $z \in \mathbb{C}$ is:

A $0$

B $2$

C $3$

D $5$

Derivation & Analysis:

By triangle inequality, $|z| + |3i - z| \ge |3i - 0| = 3$. The minimum value is $3$.

Q10

MCQ 10: If $\left(\frac{1+i}{1-i}\right)^x = 1$, then the least positive integral value of $x$ is:

A $1$

B $2$

C $4$

D $8$

Derivation & Analysis:

$\frac{1+i}{1-i} = i$. Thus $i^x = 1$. The least positive integer for which $i^x = 1$ is $x = 4$.

Common questions

The argument function is multi-valued due to the periodicity of trigonometric functions ($2\pi$). Restricting it to a principal interval ensures uniqueness in polar representation.

Sign errors during quadrant placement while finding principal arguments and mishandling branch cuts when applying fractional powers.

Memorize geometric interpretations of standard equations (e.g., perpendicular bisectors, circles, Apollonius circles) rather than expanding into Cartesian coordinates algebraically.

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