Britannica Sources used in this answer

How does Euler’s formula connect complex exponentials with sine and cosine?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

Euler’s formula, expressed as $e^{i θ} = \cos (θ) + i \sin (θ)$ (where $i = \sqrt{- 1}$ ), provides a fundamental bridge between complex exponentials and trigonometric functions.21

By comparing the Taylor series for both sides of the equation, Leonhard Euler demonstrated that the sine and cosine functions are essentially components of the exponential function. This relationship allows for the fusion of two distinct real functions into a single "complex" function.2

Visually, the formula can be understood using the unit circle: as an angle $θ$ increases, $\cos (θ)$ represents the horizontal coordinate and $\sin (θ)$ represents the vertical coordinate of a point on the circle. The expression $e^{i θ}$ effectively combines these coordinates into a single complex number, showing how trigonometric functions parameterize a circle. This insight was a significant development in mathematics, illustrating how disparate mathematical concepts can be unified through complex analysis.12