## Definitions

### General

The inverse sine function, in modern notation written as arcsin(x), gives the angle θ, so that:

\[\sin \theta = x \]

Because the values of sine function range between -1 and 1, the domain of argument x, in arcsin function, is restricted to the same range: [-1,1]. Also, due to the periodical nature of the sine function, there are many angles θ that can give the same sine value (i.e. θ+2π, θ+4π, etc.). As a result, it is impossible to define a single inverse function, unless the range of the return values is restricted, so that a one-to-one relationship between θ and sinθ can be established. Therefore, multiple branches of the arcsin function can be defined. Commonly, the desired range of θ values spans between -π/2 and π/2. The branch of arcsin, in that case, is called the principal branch.

### Series

The arcsin function can be defined in a Taylor series form, like this:

\[ \begin{split} \arcsin x & = \sum_{n=0}^{\infty}\frac{\binom{2n}{n}x^{2n+1}}{4^n \left(2n+1\right)} = \\ & = x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} \cdots \end{split} \]

The above series is valid for |x|≤1. From the expanded form of the series, it can be seen that the higher terms become insignificant, for values of x close to zero, resulting in the following quite useful approximation:

\[ \arcsin x \approx x, \quad x \to 0, \quad \textrm{result x in radians} \]

### Properties

The derivative of the arcsin function is:

\[ \left(\arcsin x\right)' = \frac{1}{\sqrt{1-x^2}}, \quad |x|\ne 1 \]

The integral of the arcsin function is given by:

\[ \int \arcsin x\, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C \]

The following properties are also valid for the arcsin function:

\[ \begin{split} & \sin (\arcsin x) &= x \\ \\ & \cos (\arcsin x) &= \sqrt{1-x^2} \\ \\ & \tan (\arcsin x) &= \frac{x}{\sqrt{1-x^2}} \\ \\ & \arcsin \left(-x\right) &=- \arcsin x \\ \\ & \arcsin x &= \frac{\pi}{2} - \arccos x \\ \\ \end{split} \]