# Euler's sine and cosine

### Euler’s formulas for sine and cosine

$\begin{aligned} j\sin \omega t &= \dfrac{1}{2} (e^{+j\omega t} - e^{-j\omega t}) \\ \\ \cos \omega t &= \dfrac{1}{2} (e^{+j\omega t} + e^{-j\omega t}) \end{aligned}$

Sine and cosine emerge from vector sum of three spinning numbers in Euler’s Formula,

The green spinning number is | $+\dfrac{1}{2} e^{+j\omega t}$ | |

The pink spinning number is | $-\dfrac{1}{2} e^{-j\omega t}$ | (used to make sine) |

The red spinning number is | $+\dfrac{1}{2} e^{-j\omega t}$ | (used to make cosine) |

Sine is the yellow dot on the imaginary axis, the vector sum of green and pink.

Cosine is the orange dot on the real axis, the vector sum of green and red.

Notice the simulation reveals the $90\degree$ phase shift between sine and cosine. Watch the dots as they pass alternately pass through the origin.

Created by Willy McAllister.

Just music, no narration. The background music is *Sunday Stroll* by Huma Huma.

Animated with d3.js, source code. The image is being computed on the fly by your device.

## Questions

can khan help with big math tests?

Khan Academy is outstanding for preparing for “big math tests”. Aren’t they all big? Look in the math subjects for your grade level or math subject (pre-algebra, algebra, etc.) There’s a Course Challenge test that will guide you right to where you should start. If you have specific topics you want to hit, enter that in the Search box.

Hello. Is “j” possibly missing from the denominator of your equation for sin omega t? Best wishes.

Richard - Great catch. It’s fixed now. Thanks for letting me know.