Circular functions — the series
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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.
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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.
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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.
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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.
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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.
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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.
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The tangent counts zigzags
Arrange the numbers 1 to n so that they go up, down, up, down. There are 1, 1, 2, 5, 16, 61, 272 such arrangements, and those are, multiplied by factorials, the Taylor coefficients of tan x and sec x — the odd lengths from the tangent, the even from the secant. Désiré André proved it in 1879 by splitting a zigzag at its largest number, and the split turns out to be the derivative of the tangent written out term by term.
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Seven times half of π, and then not
The area under sin x / x, out to infinity, is π/2. Multiply in sin(x/3)/(x/3) and the area is still π/2; multiply in the waves for 5, 7, 9, 11 and 13 and it is still π/2, exactly. Add the wave for 15 and the area falls short — by about 2.3 × 10⁻¹¹. Nothing about the curves changes at the eighth factor. What changes is that 1/3 + 1/5 + … + 1/15 has just passed 1.
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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.