Page 114 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Note that the range of each of the sine and cosine, is [–1, 1]. In terms of the real functions sine and
cosine, the other four trigonometric functions can be defined as follows:
(i) A function f : S  R defined by

sin x 
f(x) = tan x = , cos x  0, " x S = R – {(2n + 1) }
cosx 2
is called the cos x Tangent Function. The range of the tangent function is ] –, +  [ = R and

the domain is S = R – {(2n + 1) }, where n is a non-negative integer.
2
(ii) A function f : S  R defined by

cosx
f(x) = cot x = , sin x  0, " x S = S – {n },
sin x
is said to be the Cotangent Function. Its range is also same as its co-domain i.e. range
= ] – ,  [= R and the domain is S = R – {n} where n is a non-negative integer.
(iii) A function f : S  R defined by

1 
f(x) = sec x = , cos x  0, " x S = S – {2n + 1) },
cosx 2
is called the Secant Function. Its range is the set

S = ] –, –1]  [1, [ and domain is S = R – {2n + 1) }.
2
(iv) A function f : S  R defined by

1
f(x) = cosec x = , sin x  0, x S = R – {n},
sin x
is called the Cosecant function. Its range is also the set S = ] –, –1]  [1, [ and domain is
S = R – { n ),

The graphs of these functions are shown in the Figure 9.4.

 
Example: Let S = [ – , ]. Show that the function f : S  R defined by
2 2
f(x) = sin x, " x S
is one-one. When is f only onto? Under what conditions f is both one-one and onto?
Solution: Recall from Unit 1 that a function f is one-one if

f(X ) = f(X )  X = X
1 2 1 2
for every x , X in the domain of f.
1 2
Therefore, here we have for any X , X  S,
1 2
f(x ) = f(x )  sin x = sin x
1 2 1 2
 sin x – sin X = 0
1 2
x  x x  x
 2 sin 1 2 cos 1 2 = 0
2 2

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