Inverse Trigonometric Function - I
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.
![\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tZ1zo9KPE1lX0mzpCbsDv23anjVEidxcCPsCw9aFCPdlutFtpltDFLD5D5n2csc6pBslEZr9FYrlYk4ijLg9hRPrpLr4EquQxCKEh2D4VSTVmyAqGAhrvXPdc-RZYG7ZmcH3PaDdjx-DLlF5kANXdvzERrZEmNIBZF9VZ-zdoJNs26WKIOnvz-LBeb_A-Tq-6xHqJwdOLr6-sGh_PMPTpVfMMR8-HV9LXCv35ma3GPYm6p3boyfbaA1Mf2f1YQ1XuqYB0rPr9m87Kpy3xR=s0-d)
2. .
![[0,\pi ]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sGKAZreFipNJYVKIZ6Zkq0censzDsjIJ1DkTWa60HOWx9oiFLdTUCU0QEpYFzudaY5YPqsMh3BVtwFoB-EjFSu5T_zWvuzTnC64lceADzUMUz77xsf7h51P-NvpD1T2BJcQMsZd8gKOzBjq3cSfc4SGfGK_1R5qS3vo4XUTIT4sTT7XCId=s0-d)
3.

4.
( arccosec x does not exist at x = 0 )
5.
(arcsec x does not exist at x = pi/2)
6.

GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS:
1.
![Domain = [ - 1,1]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sjywUKkK0SQFM5Die-nEiYOxtNGEv4Vnj8cmyPMgyiAMcp-HY7SicADm9SOhsB9_ATkMFI8je5lzVX9KOYpjlPlblLDN7wU5j60tPy1EGn0wC_NFozKDHD1u1herRGqfA_zOM2Iw-xl9g8RpRuaq4ULEn2Z2oq7hmsthH9IM2ggrRIwOiPVAwPta_6MSHl=s0-d)
![Range = \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s5u3oej-aeTi8Ts-xPgEYtAWZ5iE7Z51OEWpQZRYEevPxilgDlzWit5K9k0zAZWFeM3KUnF4qqd0BR9JuiUYTre3wlHJcdP_PqK9XTRXzP39Yiy2o6ZmLDlLaK2aKc8DzIICdpFsfqRm3kaMr7mUJDZ0URkxw62YTPgJIIfW7jVNiwSHTWiQAM1r92BsHh9KkOwayxNYlSIK1iMvt9INk0Rh8CGC8tKouGE9IXvWCGKXOxESuS59FuZVKo5x17BfY4iZKj6uJyQSDzjDsx9rg3jgAzxZmULA=s0-d)
You can also observe from graph,

2.
![Domain = [ - 1,1]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sjywUKkK0SQFM5Die-nEiYOxtNGEv4Vnj8cmyPMgyiAMcp-HY7SicADm9SOhsB9_ATkMFI8je5lzVX9KOYpjlPlblLDN7wU5j60tPy1EGn0wC_NFozKDHD1u1herRGqfA_zOM2Iw-xl9g8RpRuaq4ULEn2Z2oq7hmsthH9IM2ggrRIwOiPVAwPta_6MSHl=s0-d)
![Range = [0,\pi ]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tFj-5mHjabYEv5FtZcQ3ZMX3PWzyyonlxomu4dzVIoYzK0VRuIlWDaZ_NPYz-SFaOGHtYu9Z3cJena-nurex14YOHTbOtqaL6mk_LYfABU4dmKrrs8c2ghj-Un310y7ZdIVfVXv4rGz3aDZih33LwQJWWIXmXSAB7Gdkg17iZ0_CrufbjCcMhWm2weG9h_Eg=s0-d)
You can also observe from the graph,

See the difference between the graph of arcsin x and arccos x.
From the above graph, you can observe

3.


From the above graph you an observe,

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.
2. .
3.
4.
5.
6.
GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS:
1.
You can also observe from graph,
2.
You can also observe from the graph,
See the difference between the graph of arcsin x and arccos x.
From the above graph, you can observe
3.
From the above graph you an observe,




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