Question for Trig Gurus

[Gerson W. Barbosa]

(12-02-2014 07:34 PM)Gerson W. Barbosa Wrote:  For the range 1/2..1, we use simmetry.

This is completely wrong (including spelling), sorry! Desirable accuracy should be granted in the range [0..sqrt(2)/2], then for the range [sqrt(2)/2..1] we use the formula

asin(x) = 90° - asin(sqrt(1 - x^2))

So the previous approximation has been adjusted for a wider range. It's somewhat less accurate, but still good for practical purposes:

asin(x) ~ 90*(2*e^(-16.01031715*e^(-1.950569752*x)) + e^(x) - e^(-x))/pi

Examples:

x asin(x)

0.00 00.00
0.05 02.87
0.10 05.74
0.15 08.63
0.20 11.54
0.25 14.48
0.30 17.46
0.35 20.48
0.40 23.57
0.45 26.74
0.50 29.99 (30.00)
0.55 33.37
0.60 36.88
0.65 40.55
0.70 44.43
0.71 45.23

http://www.wolframalpha.com/input/?i=plo...282%29%2F2

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Somewhat better:

asin(x) ~ 180/pi*(0.000005592*(x + 1)^15.096 + sinh(x))

or

asin(x) ~ 90/pi*e^(-x)*(0.000011184*e^x*(x + 1)^15.096 + e^(2*x) - 1)

Examples:

x asin(x)

0.00 00.000
0.05 02.867 (02.866)
0.10 05.740 (05.739)
0.15 08.629 (08.627)
0.20 11.541 (11.537)
0.25 14.483 (14.478)
0.30 17.465 (17.458)
0.35 20.495 (20.487)
0.40 23.586 (23.578)
0.45 26.750 (26.744)
0.50 30.002 (30.000)
0.55 33.365 (33.367)
0.60 36.864 (36.870)
0.65 40.536 (40.542)
0.70 44.429 (44.427)
0.71 45.239 (45.235)

http://www.wolframalpha.com/input/?i=plo...282%29%2F2

---------------

Less accurate, but more simple:

asin(x) ~ sinh(x)*(2 - (1 - x^6)^(1/6))*180/pi