#readwise # Euler's Formula  ## Metadata - Author: [[brilliant.org]] - Full Title: Euler's Formula - URL: https://brilliant.org/wiki/eulers-formula/ ## Summary Euler's formula connects exponential functions with trigonometric functions for complex numbers. It states that $e^{ix} = \cos{x} + i\sin{x}$. A famous result from this formula is Euler's identity: $e^{i\pi} + 1 = 0$. The formula also helps express sine and cosine in terms of exponential functions. ## Highlights In [complex analysis](https://brilliant.org/wiki/complex-analysis/), Euler's formula provides a fundamental bridge between the [exponential function](https://brilliant.org/wiki/exponential-functions/) and the [trigonometric functions](https://brilliant.org/wiki/basic-trigonometric-functions/). For [complex numbers](https://brilliant.org/wiki/complex-numbers/) $x$, Euler's formula says that $e^{ix} = \cos{x} + i \sin{x}$ In addition to its role as a fundamental mathematical result, Euler's formula has numerous applications in physics and engineering. ([View Highlight](https://read.readwise.io/read/01jfjw642ycc8953ktva9f0brw)) --- Euler's formula allows for any complex number $x$ to be represented as $e^{ix}$, which sits on a unit circle with real and imaginary components $\cos{x}$ $\sin{x}$, respectively. Various operations (such as finding the roots of unity) can then be viewed as rotations along the [unit circle](https://brilliant.org/wiki/unit-circle-basic-concept-for-higher-trigonometry/).  ([View Highlight](https://read.readwise.io/read/01jfjw6qdxr0vbkabmys9yze2k)) ^lp32oo --- One immediate application of Euler's formula is to extend the definition of the trigonometric functions to allow for arguments that extend the range of the functions beyond what is allowed under the [real numbers](https://brilliant.org/wiki/real-numbers/). ^ptfdbg A couple useful results to have at hand are the facts that $e^{-ix} = \cos{x} - i \sin{x}$ so $e^{ix} + e^{-ix} = 2 \cos{x}$ It follows that $\cos{x} = \frac{e^{ix} + e^{-ix}}{2}$ and similarly $\sin{x} = \frac{e^{ix} - e^{-ix}}{2i}$​ and $\tan{x} = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}$ ([View Highlight](https://read.readwise.io/read/01jfjw7pb1bsyfn0zbhj5xrwew)) ^bwte51 ---