Because given awk atan2, it's relatively straightforward to "draw a simple triangle and derive the other inverse trig functions":
echo -1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 |
tr ' ' '\n' |
awk '{print sin($1)}' |
awk '{print atan2($1,sqrt(1-$1^2))}'
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