# 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 $OB$.

### 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

• Results about the cosecant function can be found here.