# Definition:Inverse Sine/Arcsine

## Definition

### Real Numbers

From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From Sine of Half-Integer Multiple of Pi:

- $\map \sin {-\dfrac {\pi} 2} = -1$

and:

- $\sin \dfrac {\pi} 2 = 1$

Therefore, let $g: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \to \closedint {-1} 1$ be the restriction of $\sin x$ to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$.

This function is called **arcsine of $x$** and is written $\arcsin x$.

Thus:

- The domain of $\arcsin x$ is $\closedint {-1} 1$
- The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

### Complex Plane

The principal branch of the complex inverse sine function is defined as:

- $\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$

where:

- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.

## Terminology

There exists the popular but misleading notation $\sin^{-1} x$, which is supposed to denote the **inverse sine function**.

However, note that as $\sin x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have a well-defined inverse.

The $\arcsin$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\sin^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Sin}^{-1}$ (with a capital $\text S$) is taken to mean the same as $\arcsin$.

However, this can also be confusing due to the visual similarity between that and the lowercase $\text s$.

In computer software packages, the notation $\operatorname {asin}$ or $\operatorname {asn}$ can sometimes be found.

Some sources hyphenate: **arc-sine.**

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**arc cosine, arc sine, arc tangent,**etc.**${}$**