A sine wave is a circle seen from the side

The standard introduction to sine is a right triangle and a mnemonic. It is the same shape of introduction as the one that presents a matrix as a box of numbers: a correct rule, delivered before the object it describes, so that the rule has to be memorised rather than understood. Opposite over hypotenuse. It is correct, it is enough to solve the exercises, and it leaves a real gap: nothing about a ratio of two sides suggests that plotting it should produce a smooth, endlessly repeating wave. Where does the wave come from? Triangles do not wave.

They do not. Circles do.

What the picture says #

Put a point on a circle of radius 1 and let it walk anticlockwise. At every moment it has an angle $\theta$, measured from the rightward direction, and a height above the centre. Call that height $\sin\theta$.

That is the whole definition. The wave is what the height does as the walk proceeds: up to 1 at the top of the circle, back to 0 at the far side, down to $-1$ at the bottom, back to 0. Then it repeats, because the point has returned to where it started and the circle has no memory.

Three properties of the sine function that have to be memorised from the triangle definition become unavoidable from this one:

It repeats with period $2\pi$. Because $2\pi$ is once around. Not because of an identity — because of a lap.

It is bounded between $-1$ and $1$. Because that is how tall a unit circle is.

It is smooth, with no corners. Because the point’s height changes continuously as it moves, and nothing in a circular walk ever jerks.

Cosine is the same walk, measured sideways instead of upward.

The relationship $\sin(\theta + \tfrac{\pi}{2}) = \cos\theta$, which is an identity to be looked up, is here just the statement that turning the circle a quarter turn swaps which axis is which.

So is the most-used identity in trigonometry. The point is on a circle of radius 1, so its coordinates satisfy $x^2 + y^2 = 1$, which is to say

$$\cos^2\theta + \sin^2\theta = 1.$$

That is the Pythagorean theorem applied to a triangle with hypotenuse 1, and it needs no separate proof once the circle is drawn. It is a restatement of the circle’s own definition.

It is worth noticing that the definition has been replaced, not extended. The circle account does not add cases to the triangle account; it starts somewhere else and recovers the triangle account as a fragment. Replacements of that kind are usually the sign of a good definition — the test being whether the new version makes the old special cases look inevitable rather than merely consistent. By that test the circle wins outright: the triangle’s restriction to acute angles stops being a rule and becomes a description of which part of the lap a triangle can reach.

What the triangle definition cannot do #

The triangle picture has a hard ceiling: a right triangle’s non-right angles are between $0°$ and $90°$. There is no triangle with an angle of $200°$. So “opposite over hypotenuse” simply does not define $\sin 200°$, and the extension to larger angles has to be bolted on afterwards with sign rules and quadrant diagrams, which is where a great deal of the confusion around trigonometry comes from.

The circle has no such ceiling. The point keeps walking. At $200°$ it is below and to the left of the centre, its height is negative, and $\sin 200°$ is that negative number — no rule required, no quadrant table, nothing to remember. Angles beyond $360°$ work too, and so do negative angles, which just mean walking the other way.

The triangle definition is a special case of the circle definition, restricted to the first quarter lap. Teaching the special case first, and the general case as a patch, is backwards, but it is very hard to dislodge.

Radians stop being arbitrary #

Measuring angles in degrees is a choice, and $360$ is a Babylonian one, chosen because it divides conveniently and roughly matches the days in a year. Nothing mathematical recommends it.

Radians measure the angle by the distance the point has actually walked along the circle. A full lap of a unit circle has circumference $2\pi$, so a full turn is $2\pi$ radians. A quarter turn is $\pi/2$, because that is how far the point travelled.

In the picture above, this makes the horizontal axis and the circle’s circumference the same ruler. The wave is literally the circle unrolled — the arc straightened out, with the height carried along. That is why the horizontal axis is marked in multiples of $\pi$: those are distances, not conventions.

It also explains an otherwise arbitrary-looking fact: the derivative of $\sin$ is $\cos$ only when the angle is in radians. In degrees an extra factor of $\pi/180$ appears from nowhere. It appears because degrees are a rescaling, and the rescaling has to be paid for somewhere. Radians are the units in which the circle’s own geometry does the accounting.

Circular motion is everywhere, so waves are #

Once sine is understood as the shadow of something going round, its appearance in physics stops being surprising and becomes almost inevitable.

A weight bouncing on a spring traces a sine wave in time — because the equation governing it, $\ddot{x} = -kx$, says the acceleration always points back toward the centre in proportion to the displacement, which is precisely the condition for the shadow of uniform circular motion. A pendulum does the same for small swings. Alternating current is called alternating for the same reason: a coil rotating in a magnetic field produces a voltage proportional to the sine of its angle, so the plot of that voltage is the coil’s rotation, seen from the side.

None of those systems contains a visible circle. The circle is in the mathematics — in the fact that a quantity whose second derivative pulls it back toward zero has no choice but to go round and round.

The pendulum is not one of them #

The list above contains a quiet cheat, and it is the standard one. A pendulum does not trace a sine wave. A pendulum swinging through a small angle traces something very close to one, and the qualification is doing more work than it appears to.

The governing equation for a pendulum is $\ddot\theta = -\tfrac{g}{L}\sin\theta$, not $\ddot\theta = -\tfrac{g}{L}\theta$. The restoring pull is proportional to the sine of the displacement rather than to the displacement itself, so the condition for circular-shadow motion is not met. It is only met after the approximation $\sin\theta \approx \theta$ is made, which is accurate to within a tenth of a percent at $5°$ and within four percent at $30°$ — and false, in a way that accumulates, at any amplitude a clockmaker would care about.

What actually happens is that the period depends on the amplitude. A pendulum released from $30°$ takes about 1.7% longer to swing than one released from rest at a whisker’s displacement, and the exact period is given by an elliptic integral with no expression in elementary functions at all. The neat result — period depends on length and gravity, not on how far it is pulled back — is a first-order approximation that has been promoted to a law by repetition.

This mattered commercially. Pendulum clocks keep worse time as their swing decays, and the seventeenth-century response was to force the bob onto a path where the period genuinely is amplitude-independent. That path is not a circle; it is a cycloid, and Huygens proved in 1659 that it is the only curve with the property. His cycloidal cheeks worked in principle and were defeated in practice by friction, but the mathematics stands: the shape that makes the sine wave honest is not the shape the picture is drawn on.

So the essay’s own thesis needs a boundary. Sine is the shadow of uniform circular motion, exactly and always. It is the behaviour of physical oscillators only when their restoring force is proportional to displacement, and most real restoring forces are proportional to displacement only in the limit of small displacement. The universality is real; it is a universality of the small.

It is worth separating this from a superficially similar claim. A bell curve also turns up all over physics and statistics, but for a completely different reason — it is what accumulating many small independent contributions produces, regardless of what those contributions look like. Sine turns up because of a shape; the normal distribution turns up because of a limit. Two different kinds of universality, and conflating them is a common way to misread both.

The property that makes it easy also makes it expensive #

The picture’s best selling point is periodicity: the wave repeats because the circle has no memory. That is an argument for the definition and, unexpectedly, the source of the hardest problem in computing sine at all.

A machine evaluates $\sin x$ by first reducing $x$ into a small interval near zero — one lap is as good as any other, so subtract off whole multiples of $2\pi$ — and then applying a short polynomial that is accurate only there. The second step is routine. The first step is where the difficulty is hiding, because $2\pi$ is irrational and the machine’s copy of it is not.

Reducing $x$ modulo $2\pi$ requires knowing $2\pi$ to enough precision that the remainder is still accurate. For a small angle almost nothing is needed. For $x = 10^{22}$, dividing by $2\pi$ produces a quotient with twenty-two significant digits, all of which must be discarded, and the sixteen digits that survive into the remainder come from far below the ones a double-precision value of $2\pi$ carries. Getting the answer right needs $\pi$ to something like sixty digits, held internally, for an input that itself is only accurate to sixteen. The constant has to be known far more precisely than either the input or the output.

This is not a contrived case. $\sin(10^{22})$ has a definite value, libraries are expected to return it, and the standard method — Payne–Hanek reduction — carries hundreds of bits of $\pi$ for exactly this purpose. Implementations that skipped it were wrong for large arguments for years, and the wrongness was invisible in testing because the results were plausible numbers in the right range.

There is a real irony in it. The circle picture makes periodicity look like the cheapest property sine has: the point comes back round, and nothing needs to be remembered. In arithmetic it is the most expensive, because which lap the input is on has to be worked out exactly before the memoryless part can begin — and the further along the wave the question is asked, the more of $\pi$ it takes to answer. A shape with no memory turns out to demand a great deal of it from whoever computes the shape.

The picture, run as an algorithm #

There is a happier consequence of the same picture, and it is the one piece of arithmetic here that takes this essay’s title completely literally.

Suppose a machine has no multiplier — a pocket calculator of the early 1970s, a small controller, the arithmetic unit of a hand-held navigation set. It cannot evaluate a polynomial. What it can do is add, subtract, and shift a binary number sideways, which multiplies or divides by two.

Rotating a point by a general angle is a multiplication. Rotating by the particular angle $\arctan 2^{-k}$ is not: the matrix is $\begin{pmatrix}1 & -2^{-k}\\ 2^{-k} & 1\end{pmatrix}$ apart from an overall scale, and applying it costs two shifts and two additions. So such a machine has a small fixed repertoire of turns available to it — $45°$, then $26.57°$, $14.04°$, $7.13°$, each roughly half the last — and nothing else whatever.

That repertoire is enough. Start the point at the right-hand side of the circle and walk it toward the target angle, at each step turning by the next available amount, in whichever direction reduces what is left. The angles $\arctan 2^{-k}$ shrink slowly enough that their total reaches $99.9°$, so every angle in the first quadrant is within range, and each step settles roughly one more bit. After sixteen steps the point is within about $10^{-5}$ of where it should be, and its two coordinates are $\cos\theta$ and $\sin\theta$ — obtained, exactly as the opening figure describes, by walking round a circle.

One correction is needed and it is free. Each step stretches the vector by $\sqrt{1 + 4^{-k}}$, so a walk of sixteen steps grows it by $1.64676\ldots$ — the same factor whatever the target angle, since the same sixteen turns are used every time and only their directions differ. Starting the point at $1/1.64676 = 0.60725$ rather than $1$ cancels it in advance, before the walk begins, at a cost of nothing.

This is the CORDIC algorithm. It is what was inside the HP-35 in 1972, most scientific calculators since, and a great deal of hardware that has never had a multiplier to spare. The triangle definition offers nothing to build an algorithm out of. The circle definition, taken at its word, is the algorithm.

What the picture cannot show #

The unrolling makes the shape of the sine curve inevitable. It does not show that the curve is smooth in the technical sense, and that distinction is where the trouble usually starts.

Nothing in the picture distinguishes a curve with a well-defined slope everywhere from one that merely looks continuous. That $\sin$ is differentiable, that its derivative is $\cos$, and that the limit $\lim_{h\to 0} \frac{\sin h}{h} = 1$ which underwrites both — none of that is visible. It is proved with an inequality squeezing the arc between two triangles, and the inequality is invisible at any scale a reader can see, because the three quantities agree to several decimal places long before the picture can separate them.

The circle also cannot show what happens for complex arguments, where $\sin$ stops being bounded by 1 and grows without limit. There is no circle to walk around any more; that behaviour only becomes visible once multiplication is understood as rotation and the exponential takes over.

The ladder from here #

Later rungs: the addition formulas, derived by rotating the circle rather than by memorising them. The derivative of sine, with the squeeze argument drawn. The other four functions, and why $\tan$ has asymptotes where the others do not — a fact about a projection running off to infinity, which connects to stereographic projection. Inverse functions and the branch problem. Spherical trigonometry, where the rules change and the Pythagorean theorem fails. Lissajous figures, from two circular motions at once. And the hyperbolic functions, which are the same construction on a hyperbola rather than a circle — the fourth of the conic sections, earning a second career.

Where this goes #

The wave being the natural output of rotation is the foundation for something considerably more ambitious: the claim that any repeating pattern, however jagged, can be built by adding up enough of these smooth circular shadows. That is Fourier’s idea, and it is the subject of building a square wave out of round ones, where the flat tops and vertical jumps of a square wave are assembled entirely from sines.

The other direction leads to the complex plane, where multiplication turns out to be rotation, and where sine and cosine stop being two functions and become the two coordinates of one. That is multiplying is turning, and it ends with Euler’s formula, which is this essay’s picture written as an equation.

A sine wave is a circle seen from the side. Almost everything else about trigonometry follows from taking that sentence literally.