Let us take some examples.

Example1: Find the principal value of sin^-1 (sin (2π/3))

Solution:

sin^-1 {sin (2π/3)} = sin^-1 {sin (π - π/3)}

                        = sin^-1 {sin (π/3)}              {since sin(π - θ) = sin θ}

                        = π/3

So, the principal value of sin^-1 (sin (2π/3)) is π/3

Example2: Find the principle value of cos^-1 {cos (7π/6)}

cos^-1 {cos (7π/6)} = cos^-1 {cos (2π - 5π/6)}

                        = cos^-1 {con (5π/6)}          {since cos(2π - θ) = cos θ}

                        = 5π/6

So, the principal value of cos^-1 (cos (7π/6)) is 5π/6

Example3: Find the principle value of sin^-1 (-√3/2)

Solutions:

Given, sin^-1 (-√3/2) = -sin^-1 (√3/2)          {{since sin(θ) = -sin θ}}

                            = -π/3  

So, the principal value of sin^-1 (-√3/2) is -π/3

In this way, we find the principal value of inverse trigonometric functions.