A verified round trip
sin(30°) = 0.5 exactly. Feeding that exact 0.5 into arcsine gives back 30° exactly — arcsin(sin(30°)) = 30°. The inverse function undoes the original function completely, recovering the exact starting angle.
Why this always works within the right range
Sine and arcsine are genuine inverses of each other, the same way squaring and square-rooting are — applying one after the other returns the original input, provided the starting angle falls within arcsine's usual output range (-90° to 90°). Outside that range, sine can still be computed (e.g. sin(150°) also equals 0.5), but arcsine will return only one of the possibly many angles that share that sine value.
The same relationship holds for cosine and tangent
cos(60°) = 0.5, and arccos(0.5) = 60° — the same clean round trip, just for cosine and arccosine instead of sine and arcsine. Tangent and arctangent behave the same way for their own valid range.
Why this matters practically
Knowing that these pairs are true inverses (not just approximately related) means a problem can be worked in either direction with confidence — if a triangle's side ratio is known, arcsine or arccosine reliably recovers the angle, and that angle's sine or cosine will reliably reproduce the original ratio.