The same wave seen from a different perspective. Drag the scene to rotate — side is sine, top is cosine, front is the circle.
A point going around a circle is two waves at once:
x = A cos θ
y = A sin θ
z = (pitch / 2π) θ
Look along x and the helix is a sine. Look along y and it’s a cosine. Same curve.
Sine and cosine are not two different waves. They are the same circular motion read from two directions.
A point runs around a circle at constant speed. Its height at each moment is sine. Its left-right position is cosine. Stretch that motion out through time and you get the black helix — a corkscrew whose shadow on the back wall is sine and whose shadow on the floor is cosine.
That's why they are always 90° apart: looking at a circle from the side versus from above is a quarter-turn. Hit the Sine view and the helix collapses into the red wave. Hit Cosine and it collapses into the blue one. Same object.
The sliders are the usual wave knobs. Amplitude is how big the circle is — louder, taller, more voltage. Speed is how fast the point goes around — frequency. Turns is how tightly the helix is wound, which is wavelength along the axis. Phase is where you start on the circle.
Anything that repeats smoothly is usually a circle in disguise. Engineers keep the helix in their head and work with the two shadows.
Wall current is a point going around a circle 60 times a second. The voltage you measure is one projection. A second wire 90° later is the other projection — that's how three-phase power and electric motors work.
A pure tone is this helix in time. Your speaker cone is the up-down shadow. FM radio, Wi-Fi, and cell signals are the same idea at millions of turns per second. Mix two close frequencies and you get beats — two helices slowly sliding past each other.
A crankshaft is literally this picture in steel. The piston only sees the vertical shadow (sine). The flywheel is the circle. That's why engines need a flywheel: circular motion is smooth; the up-down shadow is not.
Planets are points on (almost) circles. Seasons, day length, moon phases, and tides are the sine and cosine shadows of those circles. Full walkthrough below.
Any messy repeating signal — speech, a JPEG, an ECG — can be rebuilt from a stack of these helices at different speeds and sizes. That's MP3, MRI, and JPEG: keep the loud circles, throw away the tiny ones.
Once you see sine and cosine as a single rotating arrow, “phase” stops being a word and becomes a direction. Radar, noise-cancelling headphones, and lock-in amplifiers all ask: which way is the arrow pointing right now?
Put the Sun at the origin. A planet is the moving point on the circle. Time is the axis going into the distance. One trip around = one year, and the black helix is that year stretched out. Amplitude is orbital radius. Speed is how fast the year goes. Mercury is a tight, fast helix; Neptune is a huge, slow one.
Real orbits are slightly squashed circles (ellipses), but the same two functions still write the position: x = a cos θ, y = a sin θ. Eccentricity just breathes the amplitude a little each lap. Kepler didn't throw this picture away — he dented it.
Earth's axis is tipped 23.4°. As the planet goes around the circle, the Sun's height in the sky over a year is the sine shadow. Solstice is the peak and the trough. Equinox is where the wave crosses zero — the quarter-turn, exactly the sine/cosine offset. Hottest day lags the solstice because oceans store heat: that's a phase shift, not a different wave.
From the ground, the Sun runs a circle once a day. Solar altitude is sine of that angle — zero at sunrise, max at noon, zero at sunset. The cosine is how far east or west it is. Same helix, much faster: ~365 turns of the day helix for every 1 turn of the year helix. That's why the visualization's Turns slider matters.
The Moon is a second, smaller, faster circle around Earth (~27 days). Phase is the angle between the Sun arrow and the Moon arrow — the phase slider in this demo. New moon: arrows aligned. First quarter: 90°, which is why you see a half-lit disk (one projection lit, the other not). Full moon: 180°.
Sea level is two helices added together: a big Moon wave and a smaller Sun wave. When those arrows point the same way, spring tide (large amplitude). When they are 90° apart, neap tide (they partly cancel). You can fake this here: amplitude = how strong each pull is, phase = how aligned they are.
Mars looks like it loops backward some nights because we watch it from our own moving circle. Two helices, two speeds. The path you see in the sky is a projection — the same trick as looking at this demo from the side versus the front. Ptolemy needed epicycles; it's just two rotating arrows.
A transfer orbit, a geosynchronous satellite, a Deep Space Network pointing angle — all “where is the rotating point right now, and what's its sine/cosine?” GPS is four of these clocks in the sky; your phone solves for position from the phase of each one. Same circle, GPS speed.
Try it: set Turns to 1 for a single year, crank Amplitude for a farther planet, and use the Sine view to watch seasons. Then add turns for days-inside-a-year, or sit in Circle view and you're looking down on the solar system.
Open the solar system modelInspired by The Math Flow · space to pause · 1–4 for views