Definition:Cotangent

Definition

Trigonometry

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

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

Thus it is seen that the cotangent is the reciprocal of the tangent.

It is also seen to be the cosine over the sine.

Analysis

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:

• $\sin x$ is the sine of $x$
• $\cos x$ is the cosine of $x$
• $\tan x$ is the tangent 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 $\cot z$ is defined as:

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

where:

• $\sin z$ is the sine of $z$
• $\cos z$ is the cosine of $z$
• $\tan z$ is the tangent of $z$

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

Also see

• Sine, cosine, tangent, secant and cosecant
• Shape of Cotangent Function