Cosecant function theory

We know that for a right-angled triangle, cosecant angle is given by the ratio of hypotenuse side to the opposite side i.e. cosecθ = hypotenuse / opposite. This shows that cosecant function provides inverse results as that of sine function and hence can be formulated as cosecθ = 1 / sinθ. Thus knowing any one of them yields results for other.

Method of graphing cosecant functions

We will see the method of graphing cosecant functions along with its properties. The figure below shows a graph plotted for cosec x versus x. It is important to note that cosecant is an odd function just like that of sine.
cosecant function graph

The important properties of cosec function are given below. This properties helps understanding cosecant function graph in a easy way.

  • Cosec function is odd function i.e. cosec(-x) = -cosec x.

  • Cosecant function is having a graph that is symmetric in nature and originating from (0,0).

  • Cosecant function has a period of two.

  • It has a vertical asymptote at some value k of x.

  • Its range is from (?, -1) to (1, +?).

  • Its domain consists of every real number except (k, k) where a vertical asymptote is found.

  • cosec x = sin(90 – y). Both the functions are called co-function of each other.

  • Cosecant function is equal to the inverse of sine function.

The table given below shows different values of cosecant function with respect to the cosecant angle. As shown, cosec 00 is not defined because sin 00 is equal to 0 and the inverse is undefined.

θ

00

300

450

600

900

cosec θ

Not defined

2

2

2/√3

1

Lets solve an example to get practical knowledge of the topic.

(1) Find sin 600 by using the cosec function:

Since sine and cosec are inverse of each other, sin 600 = 1 / cosec 600 = 1 / (2/√3) = √3/2.

(2) Find the value of cosec(-300):

Since cosecant is an add function, cosec(-x) = -cosec x. Hence cosec(-300) = - cosec(300) = -2.

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