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`arccos`

Inverse cosine function

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

Syntax

```arccos(`x`)
```

Description

`arccos(x)` represents the inverse of the cosine function.

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

`arccos` 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 cosine function is multivalued. The MuPAD® `arccos` function returns the value on the main branch. The branch cuts are the real intervals (- ∞, - 1) and (1, ∞). Thus, `arccos` returns values, such that y = arccos(x) satisfies $0\le \Re \left(y\right)\le \pi$ for any finite complex x.

The `cos` function returns explicit values for arguments that are certain rational multiples of π. For these values, `arccos` 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, `arccos` is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

Examples

Example 1

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

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

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

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

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

Floating-point intervals are computed for interval arguments:

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

Example 2

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

`arccos(2*I), arccos(-I/2), arccos(-3*I)`

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

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

Example 3

Some special values are implemented:

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

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

`cos(9/10*PI), arccos(cos(9/10*PI))`

Example 4

The values jump when crossing a branch cut:

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

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

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

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

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

Example 5

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

`rewrite(arccos(x), ln)`

Example 6

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

`diff(arccos(x), x), float(arccos(3)*arctan(5 + I))`

`limit(arccos(1 + sin(x)/x), x = 0)`

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

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

Parameters

 `x`

Return Values

Arithmetical expression or floating-point interval.

`x`