Why sin and cos chase each other

2026-05-06

A trigonometry textbook will tell you that sin θ is opposite over hypotenuse. Don't think about that yet. Think about a point going round a circle.

Drag the angle. The point P walks around the unit circle. Watch its height and its horizontal extent.

The blue bar is the height of P. The orange bar is how far P sits to the right of the centre. As θ grows, both bars stretch and squash, and the panel on the right records what they did.

That's it. That's the definition.

sin θ is the height of P.
cos θ is the width of P.

Why two waves, not one

Watch them write. They're the same shape - a smooth bump that rises, peaks, falls, dips, climbs back. The blue sin wave starts at zero and climbs. The orange cos wave starts at one and falls. They are not two ideas. They are one wave, sampled a quarter-turn apart.

When P is on the right edge of the circle (θ = 0), its height is zero and its width is one. A quarter-turn later (θ = π⁄2), P is at the top: height is one, width is zero. They've swapped. Another quarter-turn and the heights have flipped sign; another, and the widths have. Round and round, forever.

So cos is just sin with a head start of π⁄2:

cos θ = sin(θ + π⁄2)

Two names for the same shape, walking a quarter-turn apart. That's the chase.

Why the circle has to be unit

We picked a circle of radius 1. If we pick radius r instead, the point's height becomes r·sin θ and its width r·cos θ - same wave, scaled. The radius is just a volume knob. The shape is everything.

And the textbook definition - opposite over hypotenuse - is hiding inside this picture. The radius from the centre to P is the hypotenuse. The blue bar is the opposite side. opposite ⁄ hypotenuse = sin θ ⁄ 1 = sin θ. The textbook was just describing this circle, with the circle taken away.

A free identity

The bars are the legs of a right triangle whose hypotenuse is the radius. Pythagoras gives you, for free:

sin² θ + cos² θ = 1

always - because the hypotenuse is always 1. The identity isn't a fact you have to memorise. It's the Pythagorean theorem wearing a different hat. Drag the slider and the sum stays at 1; it has nowhere else to go.

A point walks around a circle. Its shadow on the wall traces a wave. Spin the wall ninety degrees and you get the other wave. That's all sin and cos ever were.