Euler’s formulas for sine and cosine

$\sin \omega t = \dfrac{1}{2} (e^{+j\omega t} - e^{-j\omega t})$

$\cos \omega t = \dfrac{1}{2} (e^{+j\omega t} + e^{-j\omega t})$

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

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

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

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

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

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

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

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

Animated with d3.js, source code. The image is not a video. It’s being computed on the fly by your device.

Created by Willy McAllister.