Why do only even values of n show up in the expansion of sin4x?

[terp.asessed]

Homework Statement

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) c_n 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...

Homework Equations

posted above

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?

 

[HallsofIvy]

Homework Statement

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

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

--> f(x) = Sigma (n= 0 to infinite) c_n 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...

Homework Equations

posted above

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?

 

[terp.asessed]

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

 

[Ray Vickson]

Homework Statement

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) c_n 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...

Homework Equations

posted above

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?

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

 

[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.