# Documentation

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# arccot

Inverse cotangent function

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

arccot(x)

## Description

arccot(x) represents the inverse of the cotangent function.

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

arccot 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 cotangent function is multivalued. The MuPAD® arccot function returns the value on the main branch. The branch cut is the interval [- i, i] on the imaginary axis. Thus, arccot returns values, such that y = arccot(x) satisfies $-\frac{\pi }{2}<\Re \left(y\right)\le \frac{\pi }{2}$ for any finite complex x.

The cot function returns explicit values for arguments that are certain rational multiples of π. For these values, arccot 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.

 Note:   MuPAD defines arccot as arccot(x) = arctan(1/x), although arccot can return an unevaluated function call and does not rewrite itself in terms of arctan. As a consequence of this definition, the real line crosses the branch cut, and arccot has a jump discontinuity at the origin.

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

## Environment Interactions

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

## Examples

### Example 1

Call arccot with the following exact and symbolic input arguments:

arccot(1), arccot(1/sqrt(2)), arccot(5 + I),
arccot(1/3), arccot(0), arccot(I/2)

arccot(-x), arccot(x + 1), arccot(1/x)

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

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

Floating-point intervals are computed for interval arguments:

arccot(-4...4), arccot(0...1)

### Example 2

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

arccot(5*I), arccot(5/4*I), arccot(-3*I)

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

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

### Example 3

Some special values are implemented:

arccot(1), arccot((5 - 2*5^(1/2))^(1/2)), arccot(3^(1/2) - 2)

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

cot(9/10*PI), arccot(cot(9/10*PI))

### Example 4

The values jump when crossing a branch cut:

arccot(0.5*I + 10^(-10)), arccot(0.5*I - 10^(-10))

On the branch cut, the values of arccot coincide with the limit "from the right" for imaginary arguments x = c*i where 0 < c < 1:

limit(arccot(0.5*I - 1/n), n = infinity);
limit(arccot(0.5*I + 1/n), n = infinity);
arccot(0.5*I)

The values coincide with the limit "from the left" for imaginary arguments x = c*i where -1 < c < 0:

limit(arccot(-0.5*I - 1/n), n = infinity);
limit(arccot(-0.5*I + 1/n), n = infinity);
arccot(-0.5*I)

### Example 5

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

rewrite(arccot(x), ln)

### Example 6

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

diff(arccot(x^2), x), float(arccos(3)*arccot(5 + I))

limit(arccot(1 - sin(x)/x), x = 0)

taylor(arccot(1/x), x = 0)

series(arccot(x), x = 0)

 x

## Return Values

Arithmetical expression or floating-point interval.

x