Definition:Cosecant

Trigonometry

 * SineCosine.png

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

The cosecant of $$\angle \theta$$ is defined as being $$\frac{\text{Hypotenuse}} {\text{Opposite}} $$.

Thus it is seen that the cosecant is the reciprocal of the sine.

Real Function
Let $$x \in \C$$ be a real number.

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


 * $$\csc x = \frac 1 {\sin x}$$

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:


 * $$\csc z = \frac 1 {\sin z}$$

where:
 * $$\sin z$$ is the sine of $$z$$.

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

Also see

 * Sine, cosine, tangent, cotangent and secant.