Inverse Trigonometric Function - I

By -
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


Comments

Post a Comment

Popular posts from this blog

Fractional Part function

By -
Image
10. Fractional Part f(x) = {x}  Every number 'x' can be written as the sum of its integer and fractional parts. x = [x] + {x} Therefore fractional part, {x} = x - [x] For example, 4.7 = 4 + 0.7 or, 0.7 = 4.7 - 4 i.e. fractional part = Number - Integral Part x = 4.7 , [x] = [4.7] = 4 and {x} = {4.7} = 0.7 Consider, {-4.7} {-4.7} = -4.7 - [-4.7] = -4.7 - (-5)  or, {-4.7} = 0.3 if the number is integer, its fractional part is obviously zero. {6} = 6 - [6] = 6 - 6 = 0 Note- fractional part is always positive and it never becomes 1. in other words, fractional part of a number is the difference between the number and its integral part. Remember,  i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part. How to draw the graph, note this, , if  because the fractional part is equal to the number if n...
Read more

Properties of Modulus & Argument: Complex Number

By -
Image
Properties of Modulus: Modulus of z is the length of vector representing z form origin to the point z. 1.  If z = x + iy then   2.   &  Since,  ( equality follows when y = 0) and   (equality follows when x = 0) IMP 3.  Since, and,  4.  ,  5.  6.  7.   holds when  i.e. both the vectors are in same direction.  holds when   i.e. both the vectors are in opposite direction In general, 8.  IMP 9.  Explanation: Since,  Therefore,  or,                                                  [Since   (? click here ) ] and  10.     ( Property of Parallelogram ) Properties of Argument: Argument is the angle between the vector representing z from ...
Read more

SIgnum , Modulus and Greatest Integer Functions

By -
Image
7. Signum function , if , if , if Domain = Signum function takes the sign of the x e.g. , Clearly the Signum Function is defined for all Real values i.e. all values of 'x' but it can attain only three values -1,0 and 1. 8. Modulus Function , if , if Domain = R Properties of Modulus: For revision, here is the quick look of the properties of modulus function 1. 2. 3. 4. 9. Greatest integer function f(x) = [x] It is defined as the largest integer less than or equal to 'x' For example, , [5] = 5 , [ -8] = -8 etc. consider 1.3 Step 1 : which integers are less than or equal to 1.3 clearly, .................... -3, -2, -1, 0 and 1 Step 2 : Among the above integers which is greatest clearly it is 1 so, [1.3] = 1 Now, consider -1.3 Step 1: which integers are less than or equal to -1.3 ............ -4, -3 and -2 Step 2 : Among the above integers which is greatest clearly -2 is the greatest so, [-1.3] = -2 c...
Read more