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CH2-Inverse Triogonometric Function

LESSON NOTES
Basic concepts
Inverse of a function ‘f’ exists, if the function is one-one and onto, i.e. bijective. Since

trigonometric functions are many-one over their domains, we restrict their domains and
co-domains in order to make them one-one and onto and then find their inverse.
For example, since the domain of sine function is the set of all real numbers
−π π
and range is the closed interval [–1, 1]. If we restrict its domain to , , then it
2 2
becomes one-one and onto with range [– 1, 1]. We denote the inverse of sine function
−1
by sin . The branch with range −π π
, called the principal value branch. When we
2 2
−1
refer to the function sin , we take it as the function whose domain is [–1, 1] and
−π π −π π
−1
range is , . We write sin : [–1, 1] → , .
2 2 2 2
The domains and ranges (principal value branches) of inverse trigonometric functions
are given below.

Function Domain Range

−1
sin [–1, 1] −π π
,
2 2
−1
cos [–1, 1] 0, π

−1
tan R ( −π π
, )
2 2

cot −1 R (0, π)

π
sec −1 R – (–1, 1) 0, π − { }
2
cosec −1 R – (–1, 1) −π π
, − {0}
2 2

Note:
−1
−1
−1
1. The symbol should not be confused with ( ) .In fact is an
angle, the value of whose sine is x, similarly for other trigonometric functions.
2. Whenever no branch of an inverse trigonometric function is mentioned, we mean
the principal value branch.
3. The value of the inverse trigonometric function which lies in the range of principal
branch is its principal value.