Arc Sine

Arc Sine

The arc sine of a number x is a real number for which the sine is \(x\).

arc sinus

The argument \(x\) of the arc sine relationship is a real number between \(-1\) and \(+1\).

The relation defined by \(y \)= arcsin(\(x\)) is not a function.


The symbol used for the arc sine of a number \(x\) is “arcsin(\(x\))” which is generally read as “arc sine of \(x\).”


In the sexagesimal system of measuring angles, we have:

  • arcsin(0,5) = 30
  • arcsin(0,5) = 180n ± 30 where n is an integer.

Because sin(30°) = \(\frac{1}{2}\), arcsin(0.5) = 30°.

The same is true for all angles of “30° ± 180°.”

Educational Note

The arc sine function is the reciprocal of the sine function defined in the interval \(\left[ -\dfrac {\pi } {2},\dfrac {\pi } {2}\right]\) which is the function \(f\) of the interval \(\left[ -\dfrac {\pi } {2},\dfrac {\pi } {2}\right]\) on the interval \(\left[ -1,1\right]\) so that \(f\left( x\right)\) is the only real number for which the sine is x.

Some authors use the notation \(\sin ^{-1}x\) to indicate the arc sine of \(x\), but this notation is already used to refer to the inverse of a number and can therefore create confusion with the inverse of a sine function called the cosecant function.

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