Inverse Trigonometric Functions


Here is a short review of the general concept of inverse functions


For a one-to-one function and its inverse
1]
         If is an element of , then is an element of , and conversely

2]
          Range of = Domain of
          Domain of = Range of

3]

           


4]
     If , then for in the domain of
     and in the domain of , and conversely

           



5]
           for in the domain of

           for in the domain of



The trigonometric functions are periodic
All of the trigonometric functions fail the horizontal line test
That is, each value in the range can be associated with infinitely many values in the domain
So no trigonometric function is a one-one function
That is, no trigonometric function has an inverse function

However, if we restrict the domain of each function so that the function is a one-one function
over the restricted domain, then we can define an inverse function over the restricted domain


                    Inverse Sine Function
The inverse sine function, denoted by or ,
is defined as the inverse
of the restricted sine function
      ,
So
     and
are equivalent to
     
where ,

In other words, the inverse sine of , or the arcsine of ,
is the number or angle ,
     ,
whose sine is


          


Sine-Inverse Sine Identities

     

     


See Examples 1 – 2, pages 504 – 505, of the textbook


Inverse Cosine Function

The inverse cosine function, denoted by or ,
is defined as the inverse
of the restricted cosine function
      ,
So
       and  

are equivalent to

     
where ,

In other words, the inverse cosine of , or the arccosine of ,
is the number or angle ,
     ,
whose cosine is


          

Cosine-Inverse Cosine Identities

     

     

See Examples 3 – 4, pages 507 – 508, of the textbook.



Inverse Tangent Function

The inverse tangent function, denoted by or ,
is defined as the inverse
of the restricted tangent function
      ,
So
        and   
are equivalent to
     
where
     
and is a real number

In other words, the inverse tangent of , or the arctangent of ,
is the number or angle ,
     ,
whose tangent is



          



Tangent-Inverse Tangent Identities

     
     


See Example 5, pages 509 – 510, of the textbook.



                                  Inverse Cotangent, Secant, and Cosecant Functions

           is  equivalent  to      where 

           is  equivalent  to       where 

           is equivalent  to        where 


                          

                                                  Domain: All real number
                                                     Range:  



                        
                                                  Domain:  
                                                    Range:   



                        
                                                  Domain:  
                                                    Range:   

                   Note: The definitions of  and are not universally agreed upon.


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                                                         begin Module 2
                                                         Trigonometric Identities and Conditional Equations