How this instrument works
Inverse sine, written asin(x) or sin⁻¹(x), answers the question 'which angle has a sine of x?' It undoes what the ordinary sine function computes, and because plain sine can only ever output a value between −1 and 1, inverse sine is only defined for inputs in that same narrow band — feeding it anything outside −1 to 1 asks for an angle that simply doesn't exist.
By convention, inverse sine returns the angle from a restricted range, −90° to 90°, even though infinitely many angles technically share the same sine value (adding a full turn, or reflecting across 90°, produces others). This restricted range is called the principal value, and it's the one every calculator and programming library returns by default, chosen so the function has exactly one well-defined output for every valid input.
Right-triangle trigonometry leans on this constantly: given the ratio of a triangle's opposite side to its hypotenuse, inverse sine recovers the angle that produced that ratio in the first place — the reverse direction of the ordinary SOH step in SOH-CAH-TOA, useful whenever a triangle's sides are already known but an angle still needs finding.
- Enter a value between −1 and 1 into the x field.
- Read asin(x): the sheet returns the corresponding angle.
- Values outside −1 to 1 have no real answer and will be flagged, since no angle's sine ever reaches beyond that range.
Worked example — asin(0.6)
asin(0.6) works out to approximately 36.87° — check the reverse direction to confirm: sin(36.87°) does indeed come back to 0.6, the two functions undoing each other exactly.
asin(0) lands on exactly 0°, the trivial starting point where sine itself is also 0. And asin(1) reaches exactly 90°, the upper edge of both sine's output range and inverse sine's own principal value range — pushed any further, there would be no valid angle left to return.
Questions
What does inverse sine mean?
It answers 'which angle produces this sine value?' — the reverse operation of ordinary sine. asin(x) = θ means sin(θ) = x, restricted to a single well-defined angle between −90° and 90°.
Why is the input restricted to between −1 and 1?
Because ordinary sine can never output a value outside that range — its whole range is bounded between −1 and 1. Asking for the angle whose sine is, say, 2 has no real answer, since no angle ever produces that.
Why does inverse sine only return angles between −90° and 90°?
Because infinitely many angles technically share any given sine value, and a function needs exactly one output per input to be well-defined. Mathematicians settled on the −90° to 90° range, called the principal value, as the standard convention every calculator follows.
How is this different from inverse cosine?
Both undo a different trig function on the same kind of input. asin(0.6) and acos(0.6) return genuinely different angles, even for the identical input value, since sine and cosine trace complementary paths around the unit circle.
Where is inverse sine actually used?
Constantly in right-triangle problems: given the ratio of the opposite side to the hypotenuse, inverse sine recovers the angle that produced it — the reverse of the basic SOH step in SOH-CAH-TOA, essential whenever a triangle's sides are known but an angle still needs solving for.