Question for Trig Gurus

[Gerson W. Barbosa]

(12-05-2014 02:48 AM)Namir Wrote:  Thank you for all these approximations and the work you put in showing the tables of results and errors!

Not so much work, just copied & pasted from Excel :-)

I had never heard of a calculator which offered sin, cos and tan but not their inverses. Thanks for sharing this information with us!
If that were my only calculator, I'd use the simplest approximation as a basis for the others (perhaps there are even more simple approximations for acos or atan, but I haven't investigated yet). As we can see in the plot, it's good enough for practical purposes. No cumbersome constants to enter (no numerical curve fitting used to obtain it, only three or four plottings). If slightly more accure results are required, just replace the 6's in formula with 5.8:

180/pi*(sinh(x)*(2 - (1 - x^5.8)^(5/29))

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 14.47 (11.48)
0.30 17.45 (17.46)
0.35 20.47 (20.49)
0.40 23.55 (23.58)
0.45 26.71 (26.74)
0.50 29.95 (30.00)
0.55 33.31 (33.37)
0.60 36.81 (33.87)
0.65 40.51 (23.58)
0.70 44.46 (44.43)
√2/2 45.05 (45.00)

max abs error: ~ 6/100 degrees

Gerson.





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

P.S.:

The maximum absolute error of the latter in the range [0..√2/2] compares to that in

asin(x) ~ (sinh(x) + (689*x^9 + 1008*x^7 + 1512*x^5)/22680)*180/pi,


which can be obtained on the HP-49G/50G this way:

'ASIN(X)'
'X'
9
TAYLR
'SINH(X)'
'X'
9
TAYLR
-
EXPAND