# Sin/Cos Wave Explorer

How a dot riding a turning wheel traces the sine and cosine waves, shown as the shadows of a spiral on a grid\.

By clever\-comet\-0b731cff

Topic: Maths

[Open the interactive Jig](https://ask.jig.com/j/EHFSDxKnTESd)

## Transcript

### 1. A wheel that draws waves

A glowing dot rides around the rim of a turning wheel\. As the wheel glides along, the dot leaves a spiral, and the spiral's shadows trace two smooth waves\. They are called sine and cosine\.

Drag the dot to turn the wheel yourself, or drag anywhere else to look around\.

### 2. Put the wheel on an x/y grid

Here is the wheel face on, sitting on a grid\. The x line measures across and the y line measures up\. The wheel's radius is exactly one grid square, so the dot is always 1 away from the centre\.

Drag the dot around the wheel and watch its x and y numbers change\.

### 3. Cosine is x, sine is y

The angle starts on the right and turns counter\-clockwise\. The blue line shows how far across the dot is: that number is the cosine\. The red line shows how high the dot is: that number is the sine\.

Straight up, at 90°, the sine is 1 and the cosine is 0\.

### 4. Unroll the turning

Now the wheel glides sideways while it turns\. The dot leaves a spiral, like a stretched spring\. One full turn of 360° makes exactly one loop of the spiral\.

### 5. Sine: the shadow on the wall

Look from the front and the spiral's shadow on the back wall rises and falls\. That smooth wave is the sine wave\. It starts at 0, climbs to 1 at 90°, and sinks to \-1 at 270°\.

### 6. Cosine: the shadow on the floor

Look from above and the shadow on the floor is a wave too\. This is the cosine wave\. It starts at its highest value, 1, because the dot begins all the way to the right\.

### 7. Same wave, a quarter turn apart

Bring the floor wave up onto the wall\. It has exactly the same shape as the sine wave, just shifted by a quarter turn, 90°\. Cosine reaches every peak a quarter turn earlier\.

The two waves cross at 45°, where both equal about 0\.71\.

### 8. Faster spin, shorter waves

Slide to spin the wheel faster as it glides\. The waves squeeze closer together, because the dot finishes more turns in the same distance\. A bigger wheel would make taller waves instead\.

### 9. Circles hiding in waves

Look straight down the spiral and it becomes a circle again\. Swings, springs, sound and household electricity all wiggle in this same circle\-made wave\.

The note A that orchestras tune to wiggles 440 times every second\.