The difference between max and min is more fundamental than the actual values of the two:
The positive side, for the max, has the advantage of going first and thus biasing toward a higher max and min:
i sin(i) x
0 0.000000 0.000000
1 0.841471 0.841471
2 0.909297 1.750768
3 0.141120 1.891888
4 -0.756802 1.135086
5 -0.958924 0.176162
6 -0.279415 -0.103254
7 0.656987 0.553733
8 0.989358 1.543091
9 0.412118 1.955209
10 -0.544021 1.411188
11 -0.999990 0.411198
12 -0.536573 -0.125375
13 0.420167 0.294792
and subsequent samples then reflect this bias;
100 -0.506366 -0.127171
101 0.452026 0.324855
102 0.994827 1.319682
103 0.622989 1.942670
104 -0.321622 1.621048
105 -0.970535 0.650513
106 -0.727143 -0.076630
107 0.184782 0.108152
108 0.926819 1.034970
109 0.816743 1.851713
110 -0.044243 1.807470
111 -0.864551 0.942919
112 -0.889996 0.052923
113 -0.097182 -0.044259
200 -0.873297 0.032700
201 -0.061890 -0.029191
202 0.806418 0.777228
203 0.933310 1.710538
204 0.202120 1.912658
205 -0.714898 1.197760
206 -0.974642 0.223118
207 -0.338305 -0.115187
208 0.609068 0.493881
209 0.996467 1.490348
210 0.467719 1.958067
211 -0.491048 1.467019
212 -0.998347 0.468672
213 -0.587771 -0.119099
300 -0.999756 0.435590
301 -0.558764 -0.123174
302 0.395953 0.272779
303 0.986633 1.259411
304 0.670207 1.929618
305 -0.262404 1.667214
306 -0.953762 0.713452
307 -0.768235 -0.054783
308 0.123603 0.068820
309 0.901801 0.970621
310 0.850888 1.821509
311 0.017672 1.839181
312 -0.831791 1.007389
313 -0.916509 0.090880
400 -0.850919 0.970558
401 -0.901775 0.068783
402 -0.123543 -0.054760
403 0.768274 0.713514
404 0.953744 1.667257
405 0.262346 1.929603
406 -0.670252 1.259352
407 -0.986623 0.272729
408 -0.395897 -0.123168
409 0.558814 0.435646
410 0.999755 1.435400
411 0.521525 1.956925
412 -0.436192 1.520733
413 -0.992876 0.527857
500 -0.467772 1.490296
501 -0.996472 0.493824
502 -0.609020 -0.115196
503 0.338362 0.223166
504 0.974655 1.197821
505 0.714855 1.912676
506 -0.202179 1.710497
507 -0.933331 0.777166
508 -0.806383 -0.029217
509 0.061950 0.032733
510 0.873327 0.906060
511 0.881770 1.787830
512 0.079518 1.867349
513 -0.795842 1.071506
600 0.044182 1.851685
601 -0.816777 1.034908
602 -0.926796 0.108112
603 -0.184722 -0.076611
604 0.727184 0.650573
605 0.970521 1.621094
606 0.321565 1.942659
607 -0.623036 1.319624
608 -0.994821 0.324803
609 -0.451972 -0.127169
610 0.506418 0.379249
611 0.999209 1.378458
612 0.573332 1.951790
613 -0.379664 1.572127
700 0.543971 1.955214
701 -0.412173 1.543041
702 -0.989367 0.553674
703 -0.656941 -0.103267
704 0.279473 0.176206
705 0.958941 1.135147
706 0.756763 1.891910
707 -0.141180 1.750731
708 -0.909323 0.841408
709 -0.841438 -0.000030
710 0.000060 0.000030
711 0.841504 0.841534
712 0.909272 1.750806
713 0.141060 1.891866
800 0.893970 1.772375
801 0.105928 1.878302
802 -0.779504 1.098798
803 -0.948263 0.150535
804 -0.245194 -0.094658
805 0.683306 0.588648
806 0.983577 1.572224
807 0.379552 1.951776
808 -0.573431 1.378345
809 -0.999204 0.379141
810 -0.506314 -0.127173
811 0.452080 0.324907
812 0.994833 1.319740
813 0.622941 1.942681
900 0.997803 1.353514
901 0.594860 1.948374
902 -0.354995 1.593379
903 -0.978469 0.614910
904 -0.702343 -0.087433
905 0.219513 0.132080
906 0.939551 1.071631
907 0.795769 1.867400
908 -0.079639 1.787761
909 -0.881827 0.905934
910 -0.873268 0.032666
911 -0.061830 -0.029164
912 0.806454 0.777290
913 0.933288 1.710578
After 100 million trials (n=100,000,000), the maximum reached was 1.95815868232315 and the minimum -0.127670960610926 for a difference of 2.08582964293407.
The closeness of the difference to 2 can be attributed to the average number of samples in a positive or negative grouping being pi (approximately, as only a finite, though large, number of samples were taken) and the area of a lobe of the sine curve being 2.
Not knowing how much farther we'd have to go to get the asymptotic value if the difference (or the actual max and min), I'd just quote the difference as being approximately 2.08583, though, if we start at 3 radians rather than zero, we do get max = 0.207390270689346 min= -1.87843937224483 for a difference of 2.08582964293417, so it looks close. And starting with i=2 radians, max= 1.11668769751522 min= -0.969141945418661 difference = 2.08582964293388.
Final answer: difference is approximately 2.08582964293.
Addendum: Starting with i=2.2, to get a different set of numbers altogether, the difference is reported as 2.08582964697012, so perhaps 2.0858296 is a better approximation, showing that less precision is deserved.
Theoretical consideration: As pi is irrational the difference between the max and min should approach the same value, although the actual min and max values would be different based on the head start provided by where you start on the sine curve.
max=0; min=30; t=0;
for i= 0:100000000
t=t+sin(i);
if i<1000
if mod(i,100)<14
fprintf('%7d %9.6f %9.6f\n',i,sin(i),t)
end
end
if t>max
max=t;
end
if t<min
min=t;
end
end
disp([max min max-min])
Max and Min biased, but constant difference between
