# Definition:Cosecant

## Definition

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cosecant** of $\angle \theta$ is defined as being $\dfrac {\text {Hypotenuse}} {\text {Opposite}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **cosecant** of $\theta$ is defined as the length of $OB$.

Hence in the first quadrant, the **cosecant** is positive.

### Real Function

Let $x \in \C$ be a real number.

The real function $\csc x$ is defined as:

- $\csc x = \dfrac 1 {\sin x}$

where $\sin x$ is the sine of $x$.

The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.

### Complex Function

Let $z \in \C$ be a complex number.

The complex function $\csc z$ is defined as:

- $\csc z = \dfrac 1 {\sin z}$

where $\sin z$ is the sine of $z$.

The definition is valid for all $z \in \C$ such that $\sin z \ne 0$.

## Also denoted as

The cosecant function:

- $\csc$

can often be seen written as:

- $\cosec$

## Linguistic Note

Like **secant**, the word **cosecant** comes from the Latin **secantus**: *that which is cutting*, the present participle of **secare**: *to cut*.

The **co-** prefix, as with similar trigonometric functions, is a reference to complementary angle: see Cosecant of Complement equals Secant.

It is pronounced with an equal emphasis on both the first and second syllables: ** co-see-kant**.

## Also see

- Definition:Sine
- Definition:Cosine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Secant Function

- Results about
**the cosecant function**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**cosecant (cosec)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**cosecant (cosec)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cosecant**