# Definition:Cotangent

## Definition

### Definition from Triangle

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

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

### 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 **cotangent** of $\theta$ is defined as the length of $AB$.

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

### Real Function

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

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

- $\cot x = \dfrac {\cos x} {\sin x} = \dfrac 1 {\tan x}$

where:

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 $\cot z$ is defined as:

- $\cot z = \dfrac {\cos z} {\sin z} = \dfrac 1 {\tan z}$

where:

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

## Linguistic Note

Like **tangent**, the word **cotangent** comes from the Latin **tangentus** *that which is touching*, the present participle of **tangere** *to touch*.

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

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

## Also see

- Shape of Cotangent Function
- Cotangent is Cosine divided by Sine
- Cotangent is Reciprocal of Tangent
- Cotangent of Complement equals Tangent

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

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

## Sources

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