# arcsin

Inverse sine function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```arcsin(`x`)
```

## Description

`arcsin(x)` represents the inverse of the sine function.

The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180o.

`arcsin` is defined for complex arguments.

Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.

If the argument is a rational multiple of `I`, the result is expressed in terms of hyperbolic functions. See Example 2.

The inverse sine function is multivalued. The MuPAD® `arcsin` function returns the value on the main branch. The branch cuts are the real intervals (- ∞, - 1) and (1, ∞). Thus, `arcsin` returns values, such that y = arcsin(x) satisfies $-\frac{\pi }{2}\le \Re \left(y\right)\le \frac{\pi }{2}$ for any finite complex x.

The `sin` function returns explicit values for arguments that are certain rational multiples of π. For these values, `arcsin` returns an appropriate rational multiple of π on the main branch. See Example 3.

The values jump when the arguments cross a branch cut. See Example 4.

The float attributes are kernel functions. Thus, floating-point evaluation is fast.

## Environment Interactions

When called with a floating-point argument, `arcsin` is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Call `arcsin` with the following exact and symbolic input arguments:

```arcsin(1), arcsin(1/sqrt(2)), arcsin(5 + I), arcsin(1/3), arcsin(I), arcsin(sqrt(2))```

`arcsin(-x), arcsin(x + 1), arcsin(1/x)`

Floating-point values are computed for floating-point arguments:

`arcsin(0.1234), arcsin(5.6 + 7.8*I), arcsin(1.0/10^20)`

Floating-point intervals are computed for interval arguments:

`arcsin(-1/2...1/2), arcsin(0...1)`

### Example 2

Arguments that are rational multiples of `I` are rewritten in terms of hyperbolic functions:

`arcsin(5*I), arcsin(5/4*I), arcsin(-3*I)`

For other complex arguments unevaluated function calls without simplifications are returned:

`arcsin(1/2^(1/2) + I), arcsin(1 - 3*I)`

### Example 3

Some special values are implemented:

`arcsin(1/sqrt(2)), arcsin((5^(1/2) - 1)/4), arcsin(3^(1/2)/2)`

Such simplifications occur for arguments that are trigonometric images of rational multiples of π:

`sin(9/10*PI), arcsin(sin(9/10*PI))`

### Example 4

The values jump when crossing a branch cut:

`arcsin(2.0 + I/10^10), arcsin(2.0 - I/10^10)`

On the branch cut, the values of `arcsin` coincide with the limit "from below" for real arguments x > 1:

```limit(arcsin(2.0 - I/n), n = infinity); limit(arcsin(2.0 + I/n), n = infinity); arcsin(2.0)```

The values coincide with the limit "from above" for real x < - 1:

```limit(arcsin(-2.0 - I/n), n = infinity); limit(arcsin(-2.0 + I/n), n = infinity); arcsin(-2.0)```

### Example 5

The inverse sine function can be rewritten in terms of the logarithm function with complex arguments:

`rewrite(arcsin(x), ln)`

### Example 6

`diff`, `float`, `limit`, or `taylor`, `series`, and other system functions handle expressions involving the inverse trigonometric functions:

`diff(arcsin(x^2), x), float(arcsin(3)*arctan(5 + I))`

`limit(arcsin(1 + sin(x)/x), x = PI)`

`taylor(arcsin(x), x = 0)`

`series(arcsin(2 + x), x, 3)`

## Parameters

 `x`

## Return Values

Arithmetical expression or floating-point interval.

`x`