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