Inverse Trigonometric Functions

In Class XI, we have studied trigonometric functions, which are defined as follows: 

We already know that, if f: x→y
f(x)= y and f is bijective, 
then there exists g: y→x
g(y)= x
g is called `"f"^-1`
So, with the help of this knowledge we will find the inverse of the trigonometric functions
1)Inverse of sin-
Sin R→[-1,1]
sin x= y

`sin: [-π/2, π/2]→[-1,1]`

`sin^(-1): [-1,1]→[-π/2, π/2],` here `[-π/2, π/2]` is the principle range 

`sin^(-1)y= x`

2) Inverse of cos-
cos: R→[-1,1]

cos: [0, π]→[-1,1]
cos (x)=y is a bijective function 
`cos^(-1): [-1,1]→[0,π]`, here `[0,π]` is the principle range
`cos^-y=x`
3) Inverse of tan-
`tan: "R"- {x:x= (2n+1)π/2,  n∈ "Z"}→"R"`

`tan: [(-π/2),  (π/2)]→"R"`

tan x= y is a bijective function

`"tan"^(-1): R→(-pi/2, pi/2)`, where `(-pi/2, pi/2)` is the principle range.

`tan^(-1)y= x`

4) Inverse of cot-
`"cot": "R"- {x:x= n pi, n∈"Z"}→R}`

`cot: (0, pi)→"R"`
cot x= y is a bijective function
`cot ^(-1):" R"→(0, pi)` , where `(0, pi)` is the principle range.
`cot :"R"→ (0, pi)`
`cot^(-1)y= x`
5) Inverse of sec-
`sec: "R"-{x:x= (2n+1) pi/2}→"R"`

`sec: [0, pi]- {pi/2}→"R"- (-1,1)`

sec x= y is bijective function

`sec^-1: "R"- (-1,1) →[0, pi]- {pi/2}`, where `[0, pi]- {pi/2}` is the principle range

6) Inverse of cosec-
`cosec: "R"- {x=n pi, n∈"Z"}→"R"-(-1,1)`

`cosec: [-pi/2, pi/2] -{0}→"R"-(-1,1)`

cosec x= y is bijective function

`cosec^-1: "R"-(-1,1)→[-pi/2, pi/2]-{0}`, where `[-pi/2, pi/2] -{0}` is the principle range.

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

As we studied in the last chapter, Functions is a special relation in which no two distinct ordered pairs have same first element i.e if y=f(x), then for one value of x we cannot have two values of y.
Also we studied that trignometric ratios behave like trignometric functions. A function must be invertible for finding it's Inverse.  In this chapter, we shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations.