# can you please explain the important concepts of inverse trignometric functions and also with their formulaes?

Grade:12

## 1 Answers

Aman Bansal
592 Points
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

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