NCERT Book Class 12 Maths Chapter 2 Inverse Trigonometric Functions

18 MATHEMATICS

Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
v Mathematics, in general, is fundamentally the science of
self-evident things. — FELIX KLEIN v
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1, exists if f is one-one and onto. There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses. In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist. In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations.
Besides, some elementary properties will also be discussed.
The inverse trigonometric functions play an important Aryabhata
(476-550 A. D.)
role in calculus for they serve to define many integrals.
The concepts of inverse trigonometric functions is also used in science and engineering.
2.2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i.e., sine : R → [– 1, 1]
cosine function, i.e., cos : R → [– 1, 1]
π
tangent function, i.e., tan : R – { x : x = (2n + 1) , n ∈ Z} → R
2
cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R
π
secant function, i.e., sec : R – { x : x = (2n + 1) , n ∈ Z} → R – (– 1, 1)
2
cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)