|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}

Fri, 17 Apr 2026 06:08:12 +0000

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.