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

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

Grade:12

1 Answers

Aman Bansal
592 Points
9 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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