How this calculator works
The sine function repeats and has a mirror symmetry within each turn. Arcsine becomes a function by choosing just the central branch, from -90° to 90°. Negative ratios give negative principal angles. The other solution in a full turn is the supplement of that principal angle, reduced to the interval from 0° to 360°. At a ratio of 1 or -1 the two branches meet, so the solution table contains one distinct angle.
Formula
y=arcsin x means sin y=x with -π/2≤y≤π/2. sin θ=x has θ=y+2kπ or θ=π-y+2kπ.
Worked example
arcsin(0.5)=30°=π/6. The equation sin θ=0.5 also has 150° in the first turn. arcsin(-1)=-90°, whose equivalent first-turn angle is 270°.
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Frequently asked questions
Why is the input limited to -1 through 1?
Sine is a coordinate on a unit circle, so its value cannot extend beyond that interval.
Why does arcsin(sin 150°) return 30°?
Arcsine selects its restricted principal branch. The two angles have the same sine but only 30° lies on that branch.
Does a negative result mean an error?
No. Negative sine ratios map to negative angles on the principal arcsine branch.