Definition:Cotangent

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Definition

Definition from Triangle

SineCosine.png

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 coordinate plane.


CotangentFirstQuadrant.png


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 $A = \left({0, 1}\right)$.

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


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


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 $\cos z \ne 0$.


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

  • Results about the cotangent function can be found here.