A point rides a circle. Its shadow draws the wave. Sine, cosine and tangent fall out of one turn.
The point still rides the same circle, but here its journey is drawn out along a third axis, time, so the path it leaves is a helix. A lamp shines on it from the front, and the shadow it throws on the wall is the sine wave. Turn the picture round the axis and watch the wave split out of the circle.
The light comes from the front, so the shadow keeps only the up-and-down part of where the point is and throws away the in-and-out part. Up-and-down on the circle is sin θ. Drawn out along time, that is the sine wave. Nothing is plotted: the wall simply shows what the lamp cannot see past.
Look down the axis and you stare straight along time: the helix lines up behind itself and collapses onto the circle at the top of the page, and the shadow flattens to a line. Turn it back and the wave splits out of the circle again. The circle and the wave are one object, seen from two places.
Put the lamp above instead and the floor would show the in-and-out part, cos θ: the same shadow, cast from a quarter turn further round. That is all "cosine leads sine by a quarter turn" means.
Below: the same thing flat, then the names. Then see what ten thousand of these waves can build: Math art →
The circle and its wave side by side, drawn point by point, with the construction lines when you want them (Guides).
Or grab the point on the circle and turn it by hand.
Turn them all off with Guides Off.
Light shining from the left onto the turning point casts a shadow, and that shadow is the sine wave. Show Shadow draws the light and the shadow it throws.
A phase shift slides a wave along the horizontal axis. Show Phase Shift moves the blue cosine wave by π/2 — a quarter turn — and it lands exactly on the gold sine wave: cos(θ + π/2) = sin(θ). The same wave, started at a different moment.
Show 90° Angle adds a second point that is always a quarter turn ahead. When the main point is at the top, where sine is largest, the second one is at the left, where cosine is zero.
A vertical line at x = 1 meets the extended radius, and the height of that meeting point is tan(θ). It is the slope of the radius, which is why tan(θ) = sin(θ)/cos(θ). At 90° and 270° cosine is zero and the tangent runs away to infinity — the ember line shooting off the chart.
Each point on the wave is born from a triangle. The angle becomes distance along the wave, and the triangle's legs become the wave's height.
Inside the circle there is a right triangle. Its three sides have names, and the three ratios are just those sides divided by each other. On a unit circle the hypotenuse is 1, so sine simply is the opposite side and cosine is the adjacent side. SOH CAH TOA is how you remember which is which.
Hypotenuse — the radius, from the centre out to the point. Always 1 here.
Opposite — the side across from the angle: how high the point is, sin θ.
Adjacent — the side along the angle: how far across the point is, cos θ.
Sine = Opposite ÷ Hypotenuse · Cosine = Adjacent ÷ Hypotenuse · Tangent = Opposite ÷ Adjacent. Past a quarter turn the sides carry signs: a side that points left or down counts as negative, and the ratios follow.