Example-1 :- Find the principal value of sin⁻¹(1/√2).
Solution :-Let sin⁻¹(1/√2) = θ. sin θ = 1/√2 sin θ = sin(π/4) {Range = [-π/2, π/2]} θ = π/4 The principal value of sin⁻¹(1/√2) = π/4
Example-2 :- Find the principal value of cot⁻¹(-1/√3).
Solution :-Let cot⁻¹(-1/√3) = θ. cot θ = -1/√3 cot θ = -cot(π/3) {Range = [0, π]} cot θ = cot(π - π/3) cot θ = cot(2π/3) θ = 2π/3 The principal value of cot⁻¹(-1/√3) = 2π/3
Example-3 :- Show that
(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x
(ii) sin⁻¹(2x√1 - x²) = 2cos⁻¹x.
(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x Let x = sin θ, then θ = sin⁻¹x L.H.S sin⁻¹(2x√1 - x²) = sin⁻¹(2 sin θ √1 - sin² θ) ∴[√1 - sin² θ = cos θ] = sin⁻¹(2 sin θ cos θ) ∴[sin 2θ = 2 sin θ cos θ] = sin⁻¹(sin 2θ) = 2θ = 2 sin⁻¹x = R.H.S (ii)sin⁻¹(2x√1 - x²) = 2cos⁻¹x. Let x = cos θ, then θ = cos⁻¹x L.H.S sin⁻¹(2x √1 - x²) = sin⁻¹(2 sin θ √1 - sin² θ) ∴[√1 - sin² θ = cos θ] = sin⁻¹(2 sin θ cos θ) ∴[sin 2θ = 2 sin θ cos θ] = sin⁻¹(sin 2θ) = 2θ = 2 cos⁻¹x = R.H.S
Example-4 :- Show that
Example-5 :- Express
, -3π/2 < x < π/2 in the simplest form.
Example-6 :- Write
, x > 1 in the simplest form.
Example-7 :- Prove that
Example-8 :- Find the value of cos(sec⁻¹x + cosec⁻¹x), | x | ≥ 1.
Solution :-cos(sec⁻¹x + cosec⁻¹x) ∴[sec⁻¹x + cosec⁻¹x = π/2] = cos(π/2) = 0