Going round a circle

Before thisread these first

Reading a formulaThe two coordinates are named with letters and the rotation rule is read back into the arithmetic it stands for, and that shorthand is built there.

After thiswhat you will be able to doRead sine and cosine as the two coordinates of a turn, rotate a point with them, and identify the distances that rotation preserves.

Questionwhat this lesson answersA point going round a circle keeps changing its two coordinates while its distance from the centre never changes. What are those two numbers, and what does turning the point do to them?

Not coveredwhat this lesson leaves outWe build the two coordinates, the identity that ties them together, and what a rotation leaves alone. We do not derive the formulas for the sine or cosine of a sum. We do not do triangles beyond the right angle inside the circle. We do not show that this rotation rule is the only one with these properties, and we do not use calculus.

Watch a point on the rim of a wheel. It rises, it falls, it comes back to where it began, and it does the same thing again on the next turn.

Two things about it are changing: how high it is, and how far across it is. One thing is not: its distance from the centre. Everything here comes from that split.

An angle is an amount of turn

Before the two changing numbers, the amount of turn itself needs a measure.

Degrees cut a full turn into 360 equal pieces. That number is a historical accident and nothing depends on it. A quarter turn is 90 degrees, half a turn is 180.

The other measure is less arbitrary. Take a circle whose radius is 1, and measure the turn by how far you have travelled along the rim. A full turn covers the whole rim, which is a distance of about ${6.283}$ . That measure is called the radian, and a full turn is ${2\pi}$ radians.

Radians are what the rest of the mathematics uses, because they make the angle and the distance travelled the same number. A quarter turn is about ${1.571}$ .

The two coordinates have names

Draw a circle of radius 1 with its centre at the origin. Start at the rightmost point and turn by some angle.

The point is now somewhere on the rim. Its sideways position is called the cosine of the angle. Its height is called the sine.

That is the definition. Not a property of sine and cosine, not something discovered about triangles: the two words name those two coordinates, and nothing else.

Some values you can read straight off the picture. At no turn at all, the point is at the right, so its sideways position is 1 and its height is 0. At a quarter turn it is at the top: sideways 0, height 1. At half a turn, sideways ${-1}$ and height 0.

The two coordinates as the point goes round

What turning both points together leaves alone

first point from centre	0.92
second point from centre	0.76
gap between them	0.97
turned by	0deg

Only the last row moves. Both points are somewhere else entirely, and every distance is exactly what it was.

Two things follow immediately, and both follow from the picture rather than from a new rule.

Neither number ever leaves the range ${-1}$ to ${1}$ , because the point never leaves the circle.

And both repeat. Turn by a full turn and the point is back where it started, so both coordinates are back to what they were.

The identity that costs nothing

The point sits at distance 1 from the centre. Drop a vertical line from it down to the horizontal axis and you have a right angle, with the two coordinates as the short sides and the radius as the long one.

Pythagoras then says the squares of the two coordinates add to the square of the radius. The radius is 1, and 1 squared is 1:

(\text{sideways})^2 + (\text{height})^2 = 1.

Check it in the island’s readout as the point sweeps. The two coordinates change constantly and their squares add to 1 the whole way round.

This looks like a discovered identity about sine and cosine. It is a consequence of putting the point on a circle of radius 1 and applying Pythagoras. Change the radius and the right-hand side changes with it.

Turning a point that is already somewhere

So far the point started at the right. Now take a point anywhere and turn it by an angle. Its new coordinates are

\begin{aligned} \text{new sideways} &= (\text{old sideways})\cos\theta - (\text{old height})\sin\theta,\\ \text{new height} &= (\text{old sideways})\sin\theta + (\text{old height})\cos\theta. \end{aligned}

Two facts about that rule matter later, and both can be watched rather than taken on faith.

Turns add. Turn by one angle, then by another, and you land exactly where a single turn by their total would have put you. That follows from what turning means: the amounts of turn simply accumulate.

Turning does not change any distance. Take two points, turn both by the same angle, and their distances from the centre are unchanged and so is the gap between them. The island’s second panel runs that: three distances sitting still while both points move.

That second fact is the one the AI arc needs. The lesson on a list of numbers as one thing writes it using the dot product, which is a different way of saying the same thing.

Where this stops

Working out the coordinates after two turns directly, without turning twice, needs formulas for the sine and cosine of a sum. Those exist and this lesson does not derive them.

Nothing here shows that this rotation rule is the only one that keeps distances fixed, and nothing here needs calculus, which is where the deeper reasons for these two functions live.

Doorswhat to read next, and why

The transformer blocknot written yetThis lesson builds coordinates for a turn, but it does not explain why token arithmetic needs a deliberate position signal to tell a sentence from a shuffle.
A list of numbers as one thingThe statement that turning two points together leaves the gap between them alone is written there using the dot product, which this lesson does not use.
Two rules fix the whole transformA rotation is one of the transforms that lesson describes, written as a table of numbers rather than as an angle.
The addition formulasWorking out the coordinates after two turns directly, rather than by turning twice, needs formulas for the sine and cosine of a sum. This lesson turns twice instead.

Symbolswhat each one means, and whether we defined it, measured it, or just started there

an angleStatus: defined: An amount of turn. Defined, and the unit is a choice. Degrees cut a full turn into 360 pieces, while radians measure the turn by how far you travel along the rim of a circle of radius one.
cosine of an angleStatus: defined: The sideways position of a point on a circle of radius one after turning by that angle. It is defined as that coordinate, not derived from anything.
sine of an angleStatus: defined: The height of the same point. Also a definition, and it is the same point, so the two always arrive together.
both coordinates staying between -1 and 1Status: defined: This follows from the point never leaving a circle of radius one, rather than being a separate rule about sine and cosine.
the squares of the two coordinates adding to oneStatus: defined: This is Pythagoras applied to a radius of one, so it follows from the setup. It looks like a discovered identity and it is a consequence of where the point was put.
turns addingStatus: defined: Turning by one angle and then another gives the same result as turning once by their total. This follows from what turning means rather than being assumed.
a rotation leaving distances aloneStatus: defined: Turning two points by the same angle changes neither distance from the centre nor the gap between them. Provable from the definitions, and demonstrated here rather than proved.
whether any real thing turns at a steady rateStatus: empirical: Whether a wheel, a signal or anything else actually goes round evenly is a claim about that thing, measured rather than settled by this arithmetic.

What these classifications mean

defined: circular by construction, true because we chose it
empirical: a measured claim about the world that could have come out otherwise
bottoms out: a primitive of the model, with nothing under it here
door: used here, explained elsewhere