# Properties of Inverse Trigonometric Functions – Problems

## Properties of Inverse Trigonometric Functions – Problems

Property I:

(i) sin¯¹ (sin θ) = θ                    for all θ ϵ [-π / 2, π / 2]

(ii) cos¯¹ (cos θ) = θ                   for all θ ϵ [0, π]

(iii) tan¯¹ (tan θ) = θ                   for all θ ϵ [-π / 2, π / 2]

(iv) cosec¯¹ (cosec θ) = θ           for all θ ϵ [-π / 2, π / 2]

(v) sec¯¹ (sec θ) = θ                   for all θ ϵ [0, π], θ ≠ π / 2

(vi) cot¯¹ (cot θ) = θ                   for all θ ϵ [0, π]

Property II:

(i) sin (sin¯¹ x) = x                 for all x ϵ [-1, 1]

(ii) cos (cos¯¹ x) = x                for all x ϵ [-1, 1]

(iii) tan (tan¯¹ x) = x                for all x ϵ R

(iv) cosec (cosec¯¹ x) = x        for all x ϵ [-∞, -1] U [1, ∞]

(v) sec (sec¯¹ x) = x                 for all x ϵ [-∞, -1] U [1, ∞]

(vi) cot (cot¯¹ x) = x                 for all x ϵ R

Example 1: Evaluate sin⁻¹ (-√3/2)

Solution: sin⁻¹ (-√3/2)

= – sin⁻¹ (√3/2)

(∵ sin⁻¹ (- θ) = – θ)

= – sin⁻¹ (sin [π/ 3])

(∵ sin⁻¹ (sinθ) = θ)

= – π/3

Example 2: Evaluate cos⁻¹ (½)

Solution: cos⁻¹ (½)

= cos⁻¹ (cos [π/ 3])

(∵ cos⁻¹ (cosθ) = θ)

= π/ 3

Example 3: Evaluate  cos⁻¹ (1/√2)

Solution: cos⁻¹ (1/√2)

= cos⁻¹ (cos [π/ 4])

(∵ cos⁻¹ (cosθ) = θ)

= π/ 4.