Exercise 3.  Use the method of Example 1.17 to establish trigonometric identities for    and  .  

Solution 3.

See text and/or instructor's solution manual.

If we let n=3 and use the binomial formula to expand the left side of De Moivre's formula, we obtain  

           

        

        

Equating the real parts we get

        ,   and
        
Equating the imaginary parts of we get

        

We are done.   

Aside.  We can let Mathematica double check our work.

The formula for  :

and the formula for  :

This solution is complements of the authors.

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