Trigonometric identities used in Fourier Analysis


NOTE: on this page, ``cos'' and ``sin'' refer to the functions with radian argument! For the degree-argument variants, the differentiation and integration formulas have different factors, but these vanish after division by T.

The facts that make Fourier analysis work are that for different speeds v and w

          /T
          |
(*1) (1/T)|  cos(vt) cos(wt) dt  ---> 0
          |
          /0
as T --> infty

and similarly for

(*2) cosvt sinwt,
(*3) sinvt sinwt, and
(*4) cosvt sinvt; whereas

          /T
          |     2
(*5) (1/T)|  cos (vt) dt  ---> 1/2
          |
          /0
as T --> infty

and the same for 

          /T
          |     2    
(*6) (1/T)|  sin (vt) dt.
          |
          /0

Each of these facts follows from a trigonometric identity.

For (*1), the identity is

cosvt coswt = (1/2)[cos(vt+wt) +cos(vt-wt)],

an easy consequence of the addition formula. Substituting this identity in the integral for (*1) gives 

 /T
 |                         1                1
2|  cos(vt) cos(wt) dt  = --- sin(vT+wT) + --- sin(vT-wT)
 |                        v+w              v-w  
 /0
which as a function of T is bounded in absolute value by the sum of the absolute values of (1/(v+w)) and (1/(v-w)) (note here how the hypothesis "v,w different" is necessary!); so then the integral is divided by T and T --> infty the limit of the quotient is zero.

For (*2) the relevant identity is

cosvt sinwt = (1/2)[sin(wt+vt) +sin(wt-vt)].

For (*3) it is

sinvt sinwt = (1/2)[cos(wt-vt) -cos(wt+vt)].

The limit for (*4) is established by noting that

                                 1 d     2                    
                  cosvt sin vt = - -- sin vt.
                                 2 dt

On the other hand, to establish (*5) it is sufficient to use the identity

cosvt cosvt = (1/2)[1+cos(2vt)],

so that

/T
|                  T    1
| cosvt cosvt dt = - + -- sin(2vT). 
|                  2   2v
/0
Since the second term is bounded in absolute value (by 1/2v), dividing by T and letting T --> infty gives the limit 1/2. In the same way, (*6) follows from the identity

sinvt sinvt = (1/2)[1-cos(2vt)].


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