Graph of "sin x" 

Domain and Range of inverse trigonometric functions 
Inv. trig. function y = f(x)
 Domain (x)
 Range (y)

y = Sin^{–1} x 
[1, 1] 
[– , ] 
y = Cos^{–1} x 
[– 1, 1] 
[0, π] 
y = Tan^{– 1} x 
R 
(– , ) 
y = Cosec^{– 1} x 
(– ∞, – 1] ∪ [1, ∞) 
[– , 0) ∪ (0, ] 
y = Sec^{– 1} x 
(– ∞, – 1] ∪ [1, ∞) 
[0, ) ∪ (, π] 
y = Cot^{– 1} x 
R 
(0, π) 
Trigonometric to Bijective Function Reduction
Consider the function f : R → [–1, 1] defined by f(x) = sin x, for all x ∈ R.
The function 'f' is a surjection but not an injection on R.
This is because f(2nπ + x) = f(x) for all n ∈ Z and x ∈ R.
i.e., for any t ∈ (–1, 1), there are infinite x ∈ R such that f(x) = t.
But, for any t ∈ (–1, 1), there exists unique x ∈ (–, ) such that f(x) = t.
(–, ) is a subinterval of a much larger length on which sine function is a bijection (fig).
⇒ The function g : (–, ) → (–1, 1) defined by g(x) = sin x for all x ∈ (–, ) is a bijection. Therefore, it has an inverse function from (– 1, 1) onto (– , ).
Denoting the function 'g' by 'sin', its inverse is denoted by g^{– 1} or Sin^{– 1} or arc sin.
Note that the first letter of an inverse trigonometric function is in the upper case.
The inverse sine function is thus defined as:
Sin^{–1} : (– 1, 1) → (–, )
defined by Sin^{– 1} x = θ ⇔ θ ∈ (– , )
and sin θ = x
It may be noted that:
(i) if x ∈ (– 1, 0), then Sin^{– 1} x ∈ (– , 0)
(ii) if x ∈ (0, 1), then Sin^{– 1} x ∈ (0, )
(iii) Sin^{– 1} 0 = 0
The inverse trigonometric functions of the other ratios i.e., cos, tan, cot, sec and cosec can be defined by taking the suitable domain.
The domains and ranges of the 6 inverse trigonometric functions are summerised in the adjacent table.