Page 11 - mathematics

Inverse of Transcendental Functions

1- Inverse of Trigonometric Functions
2- Inverse of Exponential Functions
3- Inverse of Hyperbolic Functions

Graph of y  sin x y y

1 1

1- Inverse of Trigonometric Functions  2  3 / 2    / 2  /2  3 /2 x
2
Since the trigonometric functions are not one-to-one, 1
so they don’t have inverse functions. However, if we
restrict their domains, then we may obtain one-to- x
one functions that have the same values as the
trigonometric functions and that have inverse over  / 2  /2
these restricted domains.
y  sin1 x 1
For example, the function y  sin x is not one –to-
one on its natural domain R. However, when the y
domain is restricted to the interval –π/2 to π/2, it
becomes one-to-one.  /2

x

1 1

 / 2

Graph of y  cos x y y

1

Important Rules  2 3 / 2   / 2  / 2  3 / 2 2 x

* y  sin1 x  sin y  x 1

1

x

 /2 

 * sin sin1 x  x , if 1  x 1 1

  y  cos1 x y
2 2 
* sin1 sin x   x , if   x   /2

1 1 x

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