CLass 12 Chapter 2 (Inverse Trigonometric Functions) Class Notes

Notes for CBSE Class 12 Mathematics Inverse Trigonometric Functions

Inverse Trigonometric Functions :

> Trigonometric functions are many-one functions but we know that inverse of function exists if the function is bijective. If we restrict the domain of trigonometric functions, then these functions become bijective and the inverse of trigonometric functions are defined within the restricted domain. The inverse of f is denoted by ' f⁻¹ '.
Let y = f(x) = sin x, then its inverse is x = sin⁻¹y.

INVERSE CIRCULAR FUNCTIONS

Function	Domain	Range
y = sin^-1x, iff x = siny	-1 ≤ x ≤ 1	\(\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]\)
y = cos^-1x, iff x = cosy	-1 ≤ x ≤ 1	\(\left[ {0,\pi } \right]\)
y = tan^-1x, iff x = tany	-∞ < x < ∞	\(\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\)
y = cot^-1x, iff x = coty	-∞ < x < ∞	\[{\left( {0,\pi } \right)}\]
y = cosec^-1x, iff x = ccosecy	\(\left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right)\)	\(\left[ { - \dfrac{\pi }{2},0} \right) \cup \left( {0,\dfrac{\pi }{2}} \right]\)
y = sec^-1x, iff x = secy	\(\left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right)\)	\(\left[ { - \dfrac{\pi }{2},0} \right) \cup \left( {0,\dfrac{\pi }{2}} \right]\)

Note :

(i) Sin⁻¹x & tan⁻¹x are increasing functions in their domain .
(ii) Cos⁻¹x & cot⁻¹x are decreasing functions in over domain.

PROPERTY - I