Read the trigonometric table around the whole circle (#743)

[Rafael-SOWNet]

Closes #743.

What was wrong

Every angle in the table of exact trigonometric values is written 2pi/n, so the table names only the angles that divide the turn evenly — and nothing at all between 2pi/5 and 2pi/3, which is most of the second quadrant and the whole lower half of the circle. Each lookup then tried a couple of identities to reach the rest, and they reach some of it and not the neighbouring rest:

sin(2pi/3)  ->  sqrt(3)/2         the table has the entry
sin(4pi/3)  ->  -sqrt(3)/2        reached by the doubled-angle identity
sin(5pi/3)  ->  sin(5/3 * pi)     the same value again, and not reached

This is how #743 was reported: the roots of unity are handed back in a + bi form, so one of the six roots of x^6 = 1 came back carrying an unevaluated sine — its own conjugate, spelled differently, two lines above it.

What the fix does

1. The half turn. sin(x) = -sin(x - pi), and the same for the cosine; the tangent's period is half a circle, so it takes no sign back. Applied once around the lookups rather than inside them, so it cannot turn back on itself and loop.

2. Table entries are read before a value is built out of the doubled angle. Both routes are exact, but the doubled-angle one builds a nested radical where the table has a flat one — cos(6pi/5) came back as -sqrt((1 + (sqrt(5) - 1)/4) / 2) for a number the table names as -(sqrt(5) + 1)/4 — so the roots of x^5 = 1 were written both ways at once. Reading the table first makes each root come out the way its conjugate does.

	was	is
sin(5/3 * pi)	sin(5/3 * pi)	-sqrt(3)/2
cos(4/5 * pi)	cos(4/5 * pi)	-(1 + sqrt(5))/4
sin(8/5 * pi)	sin(8/5 * pi)	-sqrt(10 + 2*sqrt(5))/4
tan(5/3 * pi)	tan(5/3 * pi)	-sqrt(3)
x^6 - 1 = 0	{ …, 1/2 + i * sin(5/3 * pi) }	all six exact
x^5 - 1 = 0	carried cos(4/5*pi) and sin(8/5*pi)	all five exact, conjugates matching

Measured

The risk in widening a lookup table is that it starts answering angles it does not know, so the check is a sweep rather than the reported cases: every k/n * pi of the first turn for eighteen denominators, 1158 calls in all, each resolved value checked against the numeric value of the call it replaced.

• 581 resolved before, 690 after — 109 angles that now have an exact value.
• No disagreement in either run. What was missing was missing rather than wrong, and the newly exact values are right.
• Suite 4979 → 4991 passed / 0 failed, F# 130/130.
• Corpus 112/117, 0 wrong / 0 error / 0 timeout, every verdict and answer byte-identical.
• work/rootcheck is 595/596 on this branch; its one incomplete case is A power of a polynomial is solved by inverting into itself, so the roots come back containing x #744, a separate solver defect fixed in Only invert an equation whose variable occurs once (#744) #745, not this one.

That sweep is the regression test, along with the reported angles, the roots of unity, and the angles the table always answered.

🤖 Generated with Claude Code