Expanding + Sigma (sum) .

1. Oct 18, 2014

terp.asessed

1. The problem statement, all variables and given/known data
Hello, I found this problem in the book I borrowed from the library, but this book does not have solutions in the back....I tried to lent the solution book but the library does not have it...so could someone help me out? The question is:

It is possible to decompose the function f(x) into components corresponding to a constant pattern plus all possible functions of the form 2pi/n with n as integer. Again, by this, supposing:

f(x) = sin2x = 1/2 + cos2x/2
--> f(x) = Sigma (n= 0 to infinite) cn cosnx...in this example, c0 = 1/2 and c2 = -1/2, where ALL other coefficients are zero.
So, based on the example, expand and find co-efficients for f(x) = sin4x by using double angle formulas, and then EXPLAIN why only even values of n show up.

I already figured out the first part of the question, and i am pretty sure I am right. But, I have no idea about the "Explain" part...

2. Relevant equations
posted above

3. The attempt at a solution
I figured out the expansion and already found co-efficients for f(x) = sin4x, which is:

f(x) = 3/8 - cos2x/2 + cos4x/8 by using double angle formula twice, sin2x and cos2x:

c0 = 3/8
c2 = -1/2
c4 = 1/8
...so I suppose all other coefficients are zero? Also, I still do not understand about "Explain why only even values of n show up?" Could someone help?

Last edited: Oct 18, 2014
2. Oct 18, 2014

HallsofIvy

Staff Emeritus
What function are you talking about? Is f(x) equal to sin2(x) or is it equal to 1/2+ cos(2x)/2?

3. Oct 18, 2014

terp.asessed

f(x) = sin2x was just an example provided in the book. What I am trying to solve is f(x) = sin4x

4. Oct 18, 2014

Ray Vickson

If you use double-angle formulas, what else could possibly occur?

5. Oct 18, 2014

terp.asessed

I got: sin4x = f(x) = 3/8 - cos2x/2 + cos4x/8....by using double angle formulas....I am having trouble as to what "Explain" part means.....exactly and why.