IIT JEE Blogs

Inverse of trigonometric ratios

We know that y = sin x means y is the value of sine of angle x if we consider domain and co-domain both as set R of real numbers. Sine ratio as seen from the fig. is many-one into function.

ü        But it is clear that if we restrict the domain to [-Π/2 , Π/2]  and range to [–1, 1], then. y = sin x is one-one onto and hence it is invertible.

So, y = sin x                             x ε [-Π/2 , Π/2] , y ε [–1, 1]

Þ  x = sin^–1 y                           y ε [–1, 1] , x ε [-Π/2 , Π/2]

ü        This value of x is called the principal value, i.e. belonging to [-Π/2 , Π/2]  and [-Π/2 , Π/2] range and it is called principal value range.

ü        The smallest numerical angle is called principal value.

ü        In general the inverse circular functions with their domain and range can be as given below:

Inverse Circular Function

Þ            sin^-1 x = θ        iff           sin θ = x, -Π/2 ≤ θ ≤ Π/2

Domain [-1,1]

Range [-Π/2 , Π/2]

Graph

Inverse Circular Function

Þ            cos^-1 x = θ   iff          cos θ = x, 0 ≤ θ ≤ Π

Domain [–1, 1]

Range [0, Π]

Graph

Inverse Circular Function

Þ            tan^-1 x = θ               iff   tan θ= x,  -Π/2< θ < Π/2

Domain (–∞, ∞)

Range (-Π/2 , Π/2)

Graph