Inverse Trigonometric Function - I

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Since none of the six trigonometric functions are one to one i.e. for many x we get the same values. 

As, sin 0 = 0, sin pi = 0, sin 2pi = 0 ..... and so on. They are not one-one rather they are periodic functions i.e. repeat its values after definite interval.

 Since, a function must be single valued i.e. one element of domain has only one image.
( to know about functions click here )

  Therefore to get the inverse trigonometric function we restrict the value of trigonometric function to its Principal Branch in which a function get all its values. The list of respective principal value corresponding to their inverse trigonometric functions are as follows. 

 MUG THIS CAREFULLY 

 S.No.             Function                       Principal Value Branch

 1.                   {\sin ^{ - 1}}x                            \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]

 2.                 . {\cos ^{ - 1}}x                            [0,\pi ]

 3.                   {\tan ^{ - 1}}x                           {\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}

 4.                   {\rm{cose}}{{\rm{c}}^{{\rm{ - 1}}}}{\rm{x}}                       \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right] - \{ 0\}   ( arccosec x does not exist at x = 0 )

 5.                   {\sec ^{ - 1}}x                             [0,\pi ] - \left\{ {\frac{\pi }{2}} \right\}    (arcsec x does not exist at x = pi/2)

 6.                   {\cot ^{ - 1}}x                             (0,\pi )
GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS:
1. f(x) = {\sin ^{ - 1}}x



Domain = [ - 1,1]

 Range = \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]

You can also observe from graph,

 {\sin ^{ - 1}}( - x) = - {\sin ^{ - 1}}x



2. f(x) = {\cos ^{ - 1}}x
Domain = [ - 1,1]

 Range = [0,\pi ]

 You can also observe from the graph,

 {\cos ^{ - 1}}( - x) = \pi - {\cos ^{ - 1}}x

See the difference between the graph of arcsin x and arccos x.

From the above graph, you can observe
{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2}


3. f(x) = {\tan ^{ - 1}}x

Domain = ( - \infty ,\infty )

 Range = \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)

 From the above graph you an observe,

 {\tan ^{ - 1}}( - x) = - {\tan ^{ - 1}}x


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