Exercise 1.  Find all values of the following.  

1 (i).   .  

Solution 1 (i).

Answer.       where  n  is an integer,  

and also       where  n  is an integer.  

Solution.   Use the Identity in (5-48)      and get:  

                      

There are two choices   

                     
and
                       

Therefore,      where  n  is an integer

and also      where  n  is an integer

                    The points   

                    and  .  

        You may wonder why the solutions look the way they do.  

Recall that    is an even function and the identity    holds for all z.  

We are done.   

Aside.  We can let Mathematica double check our work.

This can be accomplished by performing the computations    

               and     .  

We are really done.   

Aside.  The principal value of    can be calculated with Mathematica.

Also, we can substitute  n = 0  into the general formula    and get  

We are really really done.

                    The point    and the principal value image point  .  

Remark.  We will study mappings with the trigonometric functions in Section 10.4 and in some of the applications that follow in Chapter 11.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell