# Arc Sine

## Arc Sine

The arc sine of a number x is a real number for which the sine is $$x$$. 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.

### Notation

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

### Examples

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.