How do you determine if arcsin(sin(-2π/7))=-2π/7?

Answer 1

in general
arcsin(sinx) = x , since the operations are inverses of each other

However, since arcsinØ has no unique answer, it is common practise to state the solution closest to zero

e.g. suppose you do arcsin(sin 200°) on your calculator
sin 200° = -.342...
and arcsin(200°)
= arcsin (-.342..) = -20° , which is correct since sin(-20) = -.342...
Also using the CAST rule, we know sin 200° = sin(-20°) or -sin20°

Since -2π/7 is in quadrant IV (appr -64°)
arcsin(sin(-2π/7)) = -2π/7

try it on your calculator, make sure you set it to radians