Cosecant function theory
We know that for a rightangled 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.
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 cofunction 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 0^{0} is not defined because sin 0^{0} is equal to 0 and the inverse is undefined.
θ 
0^{0} 
30^{0} 
45^{0} 
60^{0} 
90^{0} 
cosec θ 
Not defined 
2 
√2 
2/√3 
1 
Lets solve an example to get practical knowledge of the topic.
(1) Find sin 60^{0} by using the cosec function:
Since sine and cosec are inverse of each other, sin 60^{0} = 1 / cosec 60^{0} = 1 / (2/√3) = √3/2.
(2) Find the value of cosec(30^{0}):
Since cosecant is an add function, cosec(x) = cosec x. Hence cosec(30^{0}) =  cosec(30^{0}) = 2.
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