# Euler's Formula Euler's formula describes the relationship between [[Trigonometric Functions]] and the complex [[The Exponential Function]]. $e^{i\theta} = \cos \theta + i\sin \theta$ It essentially **claims that for complex inputs the exponential function is periodic with a period of $2\pi i$ i.e. that all of its values sit on the unit circle of the complex plane**. In practical terms, ![[Readwise/Articles/Euler's Formula#^lp32oo]] Although $e^x$ sounds like repeated multiplication of $e$, because $x$ is not constrained to real numbers it actually represents something else, namely the shorthand for [[The Exponential Function]]. **Although the exponential function and repeated multiplication of $e$ are the same functions for real inputs, this is not the case for imaginary inputs, as the repeated multiplication of $e$ $i$ times makes no sense**. As such, when presented with $e^i$, one should think of the exponential function, not repeated multiplication.