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.