

                      REVIEW OF TRIG. IDENTITIES

You will need to memorize the trig. identities:

            2         2        
     (1) sin x  +  cos x  =  1,

     (2) sin(x + y)  =  (sin x)(cos y)  +  (sin y)(cos x),
 
     (3) cos(x + y)  =  (cos x)(cos y)  -  (sin x)(sin y).


You will want to learn how to derive the rest from these.  For example

                       2                 2         2
division of (1) by  cos x  gives  1 + tan x  =  sec x  and division by

   2                    2         2
sin (x)  yields  1 + cot x  =  csc x.  Using the odd and even properties  

of sine and cosine, we get, on putting  -y  for  y  in (2) and (3), that

     (4) sin(x - y)  =  (sin x)(cos y)  -  (sin y)(cos x)   and

     (5) cos(x - y)  =  (cos x)(cos y)  +  (sin x)(sin y).

Addition of (2) and (4), and addition of (3) and (5) give respectively

                           1
        (sin x)(cos y)  =  ─ [sin(x + y)  +  sin(x - y)] 
                           2
and
                           1
        (cos x)(cos y)  =  ─ [cos(x + y)  +  cos(x - y)] .
                           2
 
An identity for the product of sine functions results from subtraction 

                                         1   
of (3) from (5) viz.  (sin x)(sin y)  =  ─ [(cos(x - y) - cos(x + y)].
                                         2
                                                tan x  +  tan y
Division of (2) by (3) yields:  tan(x + y)  = ───────────────────  .
                                               1 - (tan x)(tan y)

                                                   2             2
To obtain identities needed for integration of  sin (x)  and  cos (x)

                                          2       2
we put  y = x  in (3) to get  cos 2x = cos x - sin x.  Then using

   2       2                        2                  2
sin x + cos x = 1  to eliminate  sin (x)  and then  cos x  gives:

      2      1                       2        1
   cos x  =  ─ (1 + cos 2x)  and  sin (x)  =  ─ (1 - cos 2x).
             2                                2

You will use these often enough that you will need to learn them.

To find any of the trig. functions at an obtuse angle Θ, use the
value of the trig. function at the related acute angle Φ and find the
appropriate sign.  For example:
 
                                                 ┌──
                            3π          π       \│ 2
                        cos ──  =  -cos ─  =  - ─── .
                             4          4        2                          


A function  f  is periodic with period  p  if  p  is the least positive
number for which  f(x + p) = f(x).



                   INVERSE FUNCTIONS AND LOGARITHMS

       
A function which is one to one has an inverse function.  To find this
inverse function start with  y = f(x), interchange  x  and y, then solve for
                                                         -1
 y  as a function of x.  This function of x will be y = f (x).
                                                             x
In algebra you studied the exponential function  y = f(x) = b , b > 1,
which is one to one and its inverse is: 
                             -1
                        y = f  (x) = log (x).
                -1                      b
Composition of f   and f yields
 
                                   y
                  y = log (x) <=> b  = x .
                         b

From this equivalence we obtain properties of logarithms as:

   1)  log (xy)   =  log (x)  +  log (y)
          b             b           b
             p
   2)  log (x )   =  p log (x)
          b               b

   3)  log (x/y)  =  log (x)  -  log (y)
          b             b           b

   4)  log (1)    =  O .
          b
