Aman Bansal
Last Activity: 12 Years ago
Daar Student,
The functions sin-1x, cos-1x, tan-1x, cot-1x, cosec-1x and sec-1x are called inverse circular or inverse trigonometric functions. sin-1x should not be confused with
(sin x)-1 which is equal to .
Each of the inverse circular functions is multivalued. To make each inverse circular function single valued we define principal values as follows. If x is positive, the
principal values of all the inverse circular functions lie between 0 and . If x is negative, the principal values of sin-1x, cosec-1x and tan-1x lie betweenand 0, and
those of cos-1x, sec-1x and cot-1x lie between and . From now onwards we take only the principalvalues.
Hence sin = x = sin-1x where (range) and x [-1, 1] (domain).
This ensures that the function = sin-1x is one-one and onto.
Similarly cosec = x = cosec-1x where and x (-, -1] [1, ).
cos = x = cos-1x where [0, ] and x [-1, 1].
sec = x = sec-1x where and x (-, -1] [1, ).
tan = x = tan-1x where and x (-, ).
cot = x = cot-1x where (0, ) and x (-, ).
The domains and ranges of inverse trigonometric (or inverse circular) functions are:
Function |
Domain |
Range |
sin-1 x |
[- 1, 1] |
|
cos-1 x |
[- 1, 1] |
[0, ] |
tan-1 x |
R |
|
cot-1 x |
R |
(0, ) |
sec-1 x |
(- , - 1] [1, ) |
|
cosec-1 x |
(- , - 1] [1, ) |
|
The graphs of = sin-1x, = cos-1x and = tan-1x are:
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Thanks
Aman Bansal
Askiitian Expert