Inverse Trigonometric Functions – Form f(f⁻¹(x))
Function of the form f(f⁻¹(x)):
where f(x) is Trigonometric Function
(i)Consider function f(x) = sin(sin⁻¹x).
Domain of the function is [-1, 1]
Also, sin (sin ⁻¹(x)) = x
sin(sin⁻¹x) = x for all x ϵ [-1, 1]
(ii)Consider function f(x) = cos(cos⁻¹x).
Domain of the function is [-1, 1]
Also, cos(cos⁻¹x) = x
cos(cos⁻¹x) = x for all x ϵ [-1, 1]
Graph of y = sin(sin⁻¹x) and y = cos(cos⁻¹x)
(iii)Consider function f(x) = tan(tan⁻¹x) = x, for all x ϵ R
(iv)Consider function f(x) = cot(cot⁻¹x) = x, for all x ϵ R
Graph of y = Tan(tan⁻¹x) and y = cot(cot⁻¹x)
(v)Consider function f(x) = cosec(cosec⁻¹x) = x, for all x ϵ (-∞, -1] \(\cup \) [1, ∞)
(vi)Consider function f(x) = sec(sec⁻¹x) = x, for all x ϵ (-∞, -1] \(\cup \) [1, ∞)
Graph of y = cosec(cosec⁻¹x) and y = sec(sec⁻¹x)