Page 118 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 9.3 Inverse Trigonometric Functions

Here we discussed inverse functions. You know that if a function is one-one and onto, then it
will have an inverse. If a function is not one-one and onto, then sometimes it is possible lo
restrict its domain in some suitable manner such that the restricted function is one-one and onto.
Let us use these ideas to define the inverse trigonometric functions. We begin with the inverse
of the sine function.
Refer to the graph of f(x) = sin x in Figure 9.8. The x-axis cuts the curve y = sin x at the points
x = 0, x = , x = 2. This shows that function f(x) = sin x is not one-one. If we restrict the domain
of f(x) = sinx to the interval [–/2, 71/21], then the function

 
f : [– , ] [–1, 1] defined by
2 2
71 71
f(x) = sin x, –  x 
21 21
is one-to-one as well as onto. Hence it will have the inverse. The inverse function is called the
inverse sine of x and is denoted as sin x. In other words,
y = sin-'x  x = sin y,
71 
where – y and –1 x  1.
21 2
Thus, we have the following definition:
Definition 11: Inverse Sine Function

 
A function g : [–1, 1]  [– , ] defined by
2 2
–x
g(x) = sin x, " x s [–1, 1]
is called the inverse sine function.

Again refer back to the graph of f(x) = cos x in Figure. You can easily see that cosine function is
also not one-one. However, if you restrict the domain of f(x) = cos x to the interval [0, ], then the
function f : [0, ]  [–1, 1] defined by
f(x) = cos 0, | x  ,
is one-one and onto. Hence it will have the inverse. The inverse function is called the inverse
cosine of x and is denoted by cos x (or by arc cos x). In other words,
–1
–1
y = cos x  x = cos y,
where 0 I y   and –1  x  I.
Thus, we have the following definition:

Definition 12:
A function g : [–1, 1]  {0, ] defined by
–1
g(x) = cos x, " x [–1, 1],
is called the inverse cosine function.

You can easily see from Figure that the tangent function, in general, is not one-one. However,
again if we restrict the domain of f(x) = tan x to the interval ]–/2, /2[, then the function.

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