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 $OB$.
Hence in the first quadrant, the cosecant is positive.
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
- Definition:Sine
- Definition:Cosine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Secant Function
- Results about the cosecant function can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cosecant