Formula based on trigonometry chapter

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, INVERSE TRIGONOMETRIC
FUNCTIONS
INVERSE TRIGONOMETRIC FUNCTIONS
If sin  = x is a trigonometrical equation, then the value of  which satisfies this equation is denoted by sin–1 x
and it is read as ‘sine inverse x’. It is called inverse function of sine. Similarly inverse functions of other
trigonometrical functions are defined. Hence inverse functions of trigonometrical functions are defined as follows-
sin–1 x =   sin  = x
cos–1 x =   cos  = x
–1
tan x =   tan  = x
–1
cot x =   cot  = x
sec–1 x =   sec  = x
–1
cosec x =   cosec  = x

DOMAIN AND RANGE OF INVERSE FUNCTIONS
As we know that in direct trigonometric functions, we are given the angle and we calculate the trigonometric ratio
(sine, cosine, etc.) or the value at that angle. Also to many values of the angle the value of trigonometric ratio is
 5 9
same e.g., tan  = 1 for   , , , etc. Inverse trigonometry deals with obtaining the angles, given the value
4 4 4
of a trigonometric ratio. In inverse trigonometry some restrictions have been imposed on the angles, and these are
based on the principle values of the angles.
 
The inverse of sine function is defined as sin–1x =  or arc sin x =  , where 1  x  1 and     e.g.,
2 2
1  5 13 1 1
 3 
sin1  and nothing else, although sin , sin etc. are also equal to , sin  2   3 only. Note that
2 6 6 6 2  
1  1 1 
sin1 x  Q   sin x  
sin x  sin x 
We list below the difinitions of all inverse trigonometric functions with their respective domains and ranges.
Function Domain Range
y = f(x) (permitted value of x) (permitted value of y)

  
(i) y = sin–1x [–1, 1]  2 , 2 
 

(ii) y = cos–1x [–1, 1] 0, 
  
(iii) y = tan–1 x  ,    , 
 2 2

(iv) y = cot–1 x  ,    0,  )
   
(v) y = sec–1 x  , 1  1,   0, 2    2 , 
   
    
(vi) y = cosec–1 x  , 1  1,    2 ,0   0, 2 
   
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