|e^{ix} + 1| = \sqrt{(\cos x + 1)^2 + \sin^2 x} = \sqrt{\cos^2 x + 2\cos x + 1 + \sin^2 x} = \sqrt{2 + 2\cos x}

February 22, 2026

$|e^{ix} + 1| = \sqrt{(\cos x + 1)^2 + \sin^2 x} = \sqrt{\cos^2 x + 2\cos x + 1 + \sin^2 x} = \sqrt{2 + 2\cos x}$

Title: Understanding |e^{ix} + 1|: A Deep Dive into Complex Exponential and Trigonometric Identities

|e^{ix} + 1| = √(2 + 2cos x): Unlocking the Beauty of Complex Numbers and Trigonometry

The expression |e^{ix} + 1| may appear abstract at first glance, but behind this elegant formula lies a powerful connection between complex analysis, trigonometry, and real-world applications. In this article, we’ll explore how this modulus evaluates to √(2 + 2cos x), uncover its geometric and algebraic interpretations, and highlight its relevance in engineering, physics, and mathematics.

What Does |e^{ix} + 1| Represent?

The symbol e^{ix} is a complex exponential rooted in Euler’s formula:

> e^{ix} = cos x + i sin x

Adding 1 gives:

> e^{ix} + 1 = (1 + cos x) + i sin x

The modulus (| |) of a complex number a + bi is defined as:

> |a + bi| = √(a² + b²)

Applying this:

> |e^{ix} + 1| = √[(1 + cos x)² + (sin x)²]

Expanding (1 + cos x)²:

= √[1 + 2cos x + cos² x + sin² x]

Using the fundamental Pythagorean identity:

> cos² x + sin² x = 1

we simplify:

= √[1 + 2cos x + 1] = √(2 + 2cos x)

Hence:

> |e^{ix} + 1| = √(2 + 2cos x)

The Geometric Insight

Equivalently:

> |e^{ix} + 1| = √(2(1 + cos x)) = √2 · √(1 + cos x)

This quantity represents the distance from the complex number e^{ix} (which lies on the unit circle in the complex plane) to the point −1 on the real axis.

The complex number 1 sits at (1, 0)
e^{ix} traces the circle of radius 1 centered at the origin
The modulus |e^{ix} + 1| measures how far e^{ix} is from −1 as x varies.

When x = 0, cos x = 1 ⇒ |e^{ix} + 1| = √(2 + 2) = √4 = 2 When x = π, cos x = −1 ⇒ |e^{ix} + 1| = √(2 − 2) = 0 — the closest point on the unit circle to −1.

Algebraic and Trigonometric Connection

Using the double-angle identity:

> 1 + cos x = 2cos²(x/2)

Substitute into our result:

> √(2 + 2cos x) = √(2 + 2·2cos²(x/2) − 2?) Wait — correction: 2cos²(x/2) = 1 + cos x ⇒ 2 + 2cos x = 2(1 + cos x) = 4cos²(x/2)

Therefore:

> |e^{ix} + 1| = √(2 + 2cos x) = √(4cos²(x/2)) = 2|cos(x/2)|

This reveals a beautiful balance: the magnitude depends on the cosine of half the angle — linking exponential decay/oscillations to harmonic symmetry.

Why This Identity Matters

1. Signal Processing and Fourier Analysis

In engineering, complex exponentials model oscillatory signals. Interpreting |e^{ix} + 1| helps compute signal amplitudes and phase differences in communication systems.

2. Phasor Representation

Electrical engineers use phasors — rotating complex numbers — where |e^{iθ} + 1| gives physical distances in alternate AC circuit elements.

3. Complex Dynamics and Control Theory

Roots of complex equations often involve e^{ix}, and understanding distances from fixed points (like −1) aids in stability analysis.

4. Geometric Intuition

Visualizing complex numbers on the plane becomes intuitive: distances from roots of unity to specific real points.

Common Challenges

Sign managing complex modulus: Remember |a + bi| = √(a² + b²), don’t confuse with complex addition.
Double-angle identities: Applying 1 + cos x = 2cos²(x/2) simplifies calculus and application.
Absolute value with cosine: √(4cos²(x/2)) = 2|cos(x/2)|, crucial for domains where x/2 crosses π/2.

Summary

The identity:

> |e^{ix} + 1| = √(2 + 2cos x) = 2|cos(x/2)|

bridges complex numbers and trigonometry, revealing deep symmetry in the complex plane. It exemplifies how exponential forms condense oscillatory behavior into compact algebraic expressions — empowering fields from physics to signal processing.