Basic Concepts of Inverse Trigonometric Functions

• sine, cosine, tangent, cotangent, secant, cosecant function

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.