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Concept

Sine — where it appears

The vertical coordinate of a point travelling round the unit circle, as a function of the angle turned through. It is the derivative of nothing simpler than cosine, and it is what a circle looks like from the side as it turns.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · Fourier series A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

analysis · Circular functions Lissajous figures for every coprime pair of frequencies up to 4. A 4 by 4 grid of Lissajous figures x = sin(pt + 0.3), y = sin(qt), with the crossing count 2pq − p − q under each and the non-coprime pairs left blank.

When two circular motions come home

Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

analysis · Circular functions Each side over the sine of the opposite angle is the diameter. A triangle inscribed in a circle, with the diameter from one corner drawn and joined to a second corner to make a right-angled triangle; the angle opposite the side at the far end of the diameter equals the triangle's own angle opposite that side.

Every side measured by one diameter

In any triangle, divide each side by the sine of the angle opposite it: the three answers are equal. That much is the law of sines, and it is usually left there. The common answer has a name — it is the diameter of the circle through the triangle's corners — and the reason is the inscribed angle once more. It is also why the sine was first a half-chord, why Ptolemy's theorem is the addition formula, and why a circle can hold infinitely many points all at rational distances from each other.

geometry · Inscribed angle A knot drawn with three sines: 5₂ as a Lissajous knot. Frequencies 2, 3, 7 with phases 0.1, 0.7, 0.3; 7 crossings from above; Alexander polynomial 2t − 3 + 2t⁻¹; determinant 7.

Three sines tie a knot

Let a point move with a different sine wave in each of the three directions of space, the three frequencies whole numbers with no common factor, and it traces a closed curve that is usually knotted. With frequencies 2, 3 and 7 the curve is the knot 5₂. The trefoil, the simplest knot of all, can never be made this way: the half-turn that shifting time by half a period performs forces a condition on the knot's polynomial that the trefoil fails.

analysis · Circular functions Walking a figure eight, and the sine it makes. Bernoulli's lemniscate with 24 equally spaced stations, and the graph of the distance from its centre against arc length walked: the lemniscatic sine, period 2ϖ ≈ 5.2441, beside an ordinary sine of the same period.

The sine of a figure eight

Walk round a circle and read the height against the distance walked, and the reading is the sine. Do the same on Bernoulli's figure eight and the reading is a new function, with an arc integral that has t⁴ where the circle's has t², a length Gauss found inside an average, two periods instead of one — and exactly the same list of equal divisions that compass and straightedge can draw.

analysis · Circular functions The sine rebuilt from its zeros. The curve sin πx and three partial products of Euler's formula, with 1, 3 and 10 pairs of factors; each matches the sine between its zeros and diverges outside them.

The sine, rebuilt from its zeros

A polynomial is a product of factors, one for each root. Euler treated the sine the same way — one factor for each place the wave crosses the axis — and the product he wrote down is correct, though the zeros alone do not justify it. Multiplied out, it hands over the sum of the reciprocal squares, π²/6, and every even power after it.

analysis · Circular functions The butterfly: wings that cut a chord equally. A circle with chord PQ, its midpoint M, two chords through M, and the wings AD and BC meeting PQ at X and Y, each 0.476 from M.

Wings that cut a chord equally

Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.

geometry · Inscribed angle Three angles with one tripled sine. roots of 4x^3 - 3x + sin3°: 0.017452406437, 0.857167300702, -0.874619707139.

The sine of one degree

Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

PeriodicityPiChordConvergenceCosineInscribed angleInvariantRadianRegular polygonSymmetryTaylor seriesUnit circle

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