# csc

Cosecant function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```csc(`x`)
```

## Description

`csc(x)` represents the cosecant function `1/sin(x)`.

The arguments have to be specified in radians, not in degrees. E.g., use π to specify an angle of 180o.

All trigonometric functions are defined for complex arguments.

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

Translations by integer multiples of π are eliminated from the argument. Further, arguments that are rational multiples of π lead to simplified results; symmetry relations are used to rewrite the result using an argument from the standard interval . Explicit expressions are returned for the following arguments:

.

Cf. Example 2.

The result is rewritten in terms of hyperbolic functions, if the argument is a rational multiple of `I`. Cf. Example 3.

The functions `expand` and `combine` implement the addition theorems for the trigonometric functions. Cf. Example 4.

The trigonometric functions do not respond to properties set via `assume`. Use `simplify` to take such properties into account. Cf. Example 4.

`csc(x)` is immediately rewritten as `1/sin(x)`. Cf. Example 5.

The inverse function is implemented by `arccsc`. Cf. Example 6.

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

## Environment Interactions

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

## Examples

### Example 1

We demonstrate some calls with exact and symbolic input data:

`sin(PI), cos(1), tan(5 + I), csc(PI/2), sec(PI/11), cot(PI/8)`

`sin(-x), cos(x + PI), tan(x^2 - 4)`

Floating point values are computed for floating-point arguments:

`sin(123.4), cos(5.6 + 7.8*I), cot(1.0/10^20)`

Floating point intervals are computed for interval arguments:

`sin(0 ... 1), cos(20 ... 30), tan(0 ... 5)`

For the functions with discontinuities, the result may be a union of intervals:

`csc(-1 ... 1), tan(1 ... 2)`

### Example 2

Some special values are implemented:

`sin(PI/10), cos(2*PI/5), tan(123/8*PI), cot(-PI/12)`

Translations by integer multiples of π are eliminated from the argument:

`sin(x + 10*PI), cos(3 - PI), tan(x + PI), cot(2 - 10^100*PI)`

All arguments that are rational multiples of π are transformed to arguments from the interval :

`sin(4/7*PI), cos(-20*PI/9), tan(123/11*PI), cot(-PI/13)`

### Example 3

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

`sin(5*I), cos(5/4*I), tan(-3*I)`

For other complex arguments, use `expand` to rewrite the result:

`sin(5*I + 2*PI/3), cos(PI/4 - 5/4*I), tan(-3*I + PI/2)`

```expand(sin(5*I + 2*PI/3)), expand(cos(5/4*I - PI/4)), expand(tan(-3*I + PI/2))```

### Example 4

The `expand` function implements the addition theorems:

`expand(sin(x + PI/2)), expand(cos(x + y))`

The `combine` function uses these theorems in the other direction, trying to rewrite products of trigonometric functions:

`combine(sin(x)*sin(y), sincos)`

The trigonometric functions do not immediately respond to properties set via `assume`:

`assume(n, Type::Integer): sin(n*PI), cos(n*PI)`

Use `simplify` to take such properties into account:

`simplify(sin(n*PI)), simplify(cos(n*PI))`

`assume(n, Type::Odd): sin(n*PI + x), simplify(sin(n*PI + x))`

`y := cos(x + n*PI) + cos(x - n*PI): y, simplify(y)`

`delete n, y:`

### Example 5

Various relations exist between the trigonometric functions:

`csc(x), sec(x)`

Use `rewrite` to obtain a representation in terms of a specific target function:

`rewrite(tan(x)*exp(2*I*x), sincos), rewrite(sin(x), cot)`

### Example 6

The inverse functions are implemented by `arcsin`, `arccos` etc.:

`sin(arcsin(x)), sin(arccos(x)), cos(arctan(x))`

Note that `arcsin(sin(x))` does not necessarily yield `x`, because `arcsin` produces values with real parts in the interval :

`arcsin(sin(3)), arcsin(sin(1.6 + I))`

### Example 7

Various system functions such as `diff`, `float`, `limit`, or `series` handle expressions involving the trigonometric functions:

`diff(sin(x^2), x), float(sin(3)*cot(5 + I))`

`limit(x*sin(x)/tan(x^2), x = 0)`

`series((tan(sin(x)) - sin(tan(x)))/sin(x^7), x = 0)`

## Parameters

 `x`

## Return Values

Arithmetical expression or a floating-point interval

`x`