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

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]

tangent function, i.e., tan : R – { x : x = (2n + 1)

2

π

, n ∈ Z} → R

cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R

secant function, i.e., sec : R – { x : x = (2n + 1)

2

π

, n ∈ Z} → R – (– 1, 1)

cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

Chapter

2

INVERSE TRIGONOMETRIC

FUNCTIONS

Aryabhata

(476-550 A. D.)

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34 MATHEMATICS

We have also learnt in Chapter 1 that if f : X→Y such that f(x) = y is one-one and

onto, then we can define a unique function g : Y→X such that g(y) = x, where

x ∈ X

and y = f(x), y ∈

Y. Here, the domain of g = range of f and the range of g = domain

of f. The function g is called the inverse of f and is denoted by f

–1

. Further, g is also

one-one and onto and inverse of g is f. Thus, g

–1

= (f

–1

)

–1

= f. We also have

(f

–1

o f ) (x) = f

–1

(f (x)) =

f

–1

(y) = x

and (f o f

–1

) (y) = f (f

–1

(y))

= f(x) = y

Since the domain of sine function is the set of all real numbers and range is the

closed interval [–1, 1]. If we restrict its domain to

,

2 2

−π π

, then it becomes one-one

and onto with range [– 1, 1]. Actually, sine function restricted to any of the intervals

−

3

2 2

π π

,

,

,

2 2

−π π

,

3

,

2 2

π π

etc., is one-one and its range is [–1, 1]. We can,

therefore, define the inverse of sine function in each of these intervals. We denote the

inverse of sine function by sin

–1

(arc sine function). Thus, sin

–1

is a function whose

domain is [– 1, 1] and range could be any of the intervals

3

,

2 2

− π −π

,

,

2 2

−π π

or

3

,

2 2

π π

, and so on. Corresponding to each such interval, we get a branch of the

function sin

–1

. The branch with range

,

2 2

−π π

is called the principal value branch,

whereas other intervals as range give different branches of sin

–1

. When we refer

to the function sin

–1

, we take it as the function whose domain is [–1, 1] and range is

,

2 2

−π π

. We write sin

–1

: [–1, 1] →

,

2 2

−π π

From the definition of the inverse functions, it follows that sin (sin

–1

x) = x

if – 1 ≤ x ≤ 1 and sin

–1

(sin x) = x if

2 2

x

π π

− ≤ ≤

. In other words, if y = sin

–1

x, then

sin y = x.

Remarks

(i) We know from Chapter 1, that if y = f(x) is an invertible function, then x = f

–1

(y).

Thus, the graph of sin

–1

function can be obtained from the graph of original

function by interchanging x and y axes, i.e., if (a, b) is a point on the graph of

sine function, then (b, a) becomes the corresponding point on the graph of inverse

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INVERSE TRIGONOMETRIC FUNCTIONS 35

of sine function. Thus, the graph of the function y = sin

–1

x can be obtained from

the graph of y = sin x by interchanging x and y axes. The graphs of y = sin x and

y = sin

–1

x are as given in Fig 2.1 (i), (ii), (iii). The dark portion of the graph of

y = sin

–1

x represent the principal value branch.

(ii) It can be shown that the graph of an inverse function can be obtained from the

corresponding graph of original function as a mirror image (i.e., reflection) along

the line y = x. This can be visualised by looking the graphs of y = sin x and

y = sin

–1

x as given in the same axes (Fig 2.1 (iii)).

Like sine function, the cosine function is a function whose domain is the set of all

real numbers and range is the set [–1, 1]. If we restrict the domain of cosine function

to [0, π], then it becomes one-one and onto with range [–1, 1]. Actually, cosine function

Fig 2.1 (ii)

Fig 2.1 (iii)

Fig 2.1 (i)

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