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# Cosecant Function and the Inverse Cosecant Function

Like sine function, the cosecant function is an odd function and has period 2π.

The Cosecant Function and the Inverse Cosecant Function

Like sine function, the cosecant function is an odd function and has period 2π . The values of cosecant function y = cosec x repeat after an interval of length 2π .Observe that y = cosec x = 1/sinx  is not defined when sin x = 0 . So, the domain of cosecant function is R\ {: n Z}. Since −1 ≤ sin x ≤ 1 , y = cosec x does not take any value in between −1 and 1. Thus, the range of cosecant function is (−∞,1] [1, ∞) .

## 1. Graph of the cosecant function

In the interval (0, 2π ) ,  the  cosecant function  is continuous everywhere except at the point  x = π. It has neither maximum nor minimum. Roughly speaking, the value of y = cosec x falls from ∞ to 1 for x (0, π/2], it raises from 1 to ∞ for x[π/2, π). Again, it raises from −∞ to -1 for x∈ (π, 3π/2]  and falls from -1 to −∞ for x [3π/2, 2 π).

The graph of y = cosec x, x (0, 2π ) \ {π } is shown  in  the  Fig.  4.19. This  portion  of  the  graph is repeated  for  the  intervals …, (−4π , − 2π ) \ {−3π }, (−2π , 0) \ {−π }, (2π , 4π ) \ {3π }, (4π , 6π ) \ {5π },…...

The entire graph of y = cosec x is  shown  in Fig. 4.20.

## 2. The inverse cosecant function

The cosecant function, cosec : [- π/2 , 0 ) U ( 0, π ] → (-∞, -1]  U [1, ∞) is bijective in the restricted domain [- π , 0 ) U ( 0, π ] . So, the inverse cosecant function is defined with the domain (-∞, -1] U [1, ∞) and the range [- π , 0 ) U ( 0, π ] .

### Definition 4.6

The inverse cosecant  function cosec-1 : (-∞, -1]  U  [1, ∞) →  [ - π /2 , 0)  U ( 0, π/2]  is defined by cosec-1  (x) = y  if and only if cosec y = x  and y [ - π/2, 0)  U ( 0, π/2].

## 3. Graph of the inverse cosecant function

The inverse cosecant function, y = cosec-1 x is a function whose domain is R \ (- 1, 1) and the range is [- π/2, π/2 ] \ {0}. That is, cosec-1 : (-∞, -1] U [1, ∞) → [- π , 0 ) U ( 0, π ] .

Fig. 4.21 and Fig. 4.22 show the graphs of cosecant function in the principal domain and the inverse cosecant function in the corresponding domain respectively.

Tags : Definition, Graph, Properties , 12th Mathematics : UNIT 4 : Inverse Trigonometric Functions
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12th Mathematics : UNIT 4 : Inverse Trigonometric Functions : Cosecant Function and the Inverse Cosecant Function | Definition, Graph, Properties