Euler's Formula in Trigonometry

Introduction to Euler's Formula

Euler's formula, named after Leonhard Euler, establishes a fundamental relationship between trigonometric functions and the complex exponential function. For any real number θ (theta), representing an angle in radians, Euler's formula states that the complex exponential function e raised to the power of iθ (where i is the imaginary unit) equals the sum of the cosine of θ with the product of i and the sine of θ.

Mathematical Background

This formula is a cornerstone in the field of complex analysis and has profound implications in various branches of mathematics and physics, including Fourier analysis and quantum mechanics. It simplifies calculations involving trigonometric and exponential functions. Euler's formula is also a special case of the more general Euler's identity, considered the most beautiful theorem in mathematics: e^(iπ) + 1 = 0.

Practical Applications:

• Analysis of waves and oscillations in physics.
• Electrical engineering, for analyzing alternating current circuits.
• Quantum mechanics, where it helps in solving Schrödinger's equation.

Parameter Usage:

• angleInRadians= angle θ in radians for which to apply Euler's formula

Summary

This calculator outputs the representation of Euler's formula for a given angle in radians, aiding in the study and application of complex analysis, physics, and engineering.