Trigonometry Review

                  Fourteen Trigonometric Identities
             
 We present here 14 trigonometric identities that every mathematics
 student should know with perfect accuracy.  Memorize some of them, and 
 know how to quickly derive the others from the ones you memorize.  

 The first four identities are basic.  

         2       2       
 1.   sin x + cos x = 1

 2.   sin(-x) = -sin(x)
 3.   cos(-x) = cos(x)
 4.   sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
 
 The next three identities follow easily from identities #2, #3, and #4.  
 To derive  #6, differentiate both sides of #4 with respect to one of 
 the variables.

 5.  sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
 6.  cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
 7.  cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

 From #4 and #6 the next two identities are easily derived.


 8.  sin(2x) = 2sin(x)cos(x)
 

                  2         2
 9.  cos(2x) = cos (x) - sin (x)


 
 From #9, the next two identities follow.
 

        1 + cos(2x)          2
 10.   -------------  =   cos (x)
            2


        1 - cos(2x)          2
 11.   -------------  =   sin (x)
            2

        
 Next come the half angle formulas, which follow from #10 and #11.  


          2             1 + cos(a)
 12.   cos (a/2)   =   ------------
                            2


          2             1 - cos(a)
 13.   sin (a/2)   =   ------------
                            2
                                                   
                                                           2
 Finally,  #14 follows by dividing both sides of #1 by  cos (x).


           2                2
 14.    tan (x) +  1  =  sec (x)

 
                 Exact values of trigonometric functions

 Every mathematics student should know the following table by heart.


     x        sin(x)       cos(x)          tan(x)            sec(x)
        
     0          0            1               0                 1

   pi/6        1/2      (sqrt(3))/2      1/(sqrt(3))      2/(sqrt(3))

   pi/4     1/(sqrt(2)   1/(sqrt(2)          1              sqrt(2)

   pi/3    (sqrt(3))/2      1/2            sqrt(3)             2

   pi/2         1            0            undefined        undefined

 


               Definitions of the Trigonometric Functions

 Each mathematics student should be able to define the trigonometric
 functions.  A concise definition follows.
 
 An angle in the coordinate plane is in standard position if its vertex 
 coincides with the origin, (0,0), and its initial side coincides with 
 the positive x-axis.  For such an angle, T, let P(a,b) be a point 
 other than (0,0) on its terminal side.  Let r be the distance from P 
 to the origin.  Note that by the Pythagorean Theorem, 

                                 2    2
                    r =  sqrt(  a  + b  )  .


 The functions sin(T) and cos(T) are now defined as follows:

                       b                                a
           sin(T)  =  ---        and        cos(T)  =  ---
                       r                                r


 The other four trigonometric functions are defined in terms of 'sin' 
 and 'cos' as follows:

                     sin(T)                       cos(T)
         tan(T)  =   ------           cot(T)  =   ------
                     cos(T)                       sin(T)


                       1                            1
         sec(T)  =   -----            csc(T)  =   -----
                     cos(T)                       sin(T)

 
 An angle is often represented by its radian measure or its measure in 
 degrees.  It is important to use radians whenever derivatives are 
 involved, because otherwise the familiar formulas for derivatives of 
 trigonometric functions are not valid.  

 When dealing with a triangle, it is customary to consider all the angles to 
 be between 0 radians and pi radians.  Two fundamental tools for solving 
 general triangles are the law of sines and the law of cosines.

 LAW OF SINES and LAW OF COSINES

   Let the lengths of the three sides of a triangle be denoted a, b,
   and c.  Let the angle opposite side a be u, and the angle opposite 
   side b be v and the angle opposite side c be w.  

                    sin u     sin v     sin w
   LAW OF SINES:   ------- = ------- = -------   
                      a         b         c

                      2       2       2
   LAW OF COSINES:   c   =   a   +   b   -  2ab cos w



  Exercises

  1.  Show that sin(pi/5) = sin(4pi/5).

  2.  Express sin(4pi/5) in terms of sin(pi/5) and cos(pi/5).

  3.  Show that cos(pi/5) is a root of the polynomial  8x^3 - 4x - 1.

  4.  Show that the only positive root of the polynomial in problem 3
      is 

                      sqrt(5) + 1
                     -------------
                           4          ,

      and conclude that this expression is cos(pi/5).

  6.  Give an expression for the exact value of sin(pi/5).

  7.  Expand sin(pi/5 + 4pi/5) to get an equation of the form:
               
                   0 = [expression in sin(pi/5) and cos(pi/5)]

  8.  Simplify the equation from problem 7, by cancelling out a common factor 
      on the right.

  9.  Get the equation from 8 into the form 0 =  [ a quadratic expression in
      either (sin(pi/5))^2  or  (cos(pi/5))^2].  

 10.  Solve that quadratic equation, and do the obvious things to find 
      expressions for sin(pi/5) and cos(pi/5).  Problems 7 - 10 constitute
      an alternate derivation of the information in problems 5 and 6.