Approximate pi to 24 digits via keyboard

[Rick314]

This works with several HP calculators I've tried it on, but not on other scientific calculators like a TI-83+ for example. In radians, for eps << 1, sin(pi - eps) = sin(pi)*cos(eps) - cos(pi)*sin(eps) = 0 - (-1)*sin(eps) = sin(eps) ~= eps. So:

sin(3.14159265358) * 1e12 = 9.79323846264, the next 12 digits of pi (3.14159265358979323846264...).

Notice this requires entering the first 12 *truncated* digits of pi (3.1415926535*8*) to be sure the value is slightly less than pi, not the 12-digit calculator *rounded approximation* to pi of 3.1415926535*9* that is actually a little larger than pi. This can be known without knowing the value of pi to begin with, because sin(<calculator approximation to pi>) is a small *negative* number. So the calculator approximation is larger than pi and the 12th truncated digit must be an 8, not a 9.