What is the Fourier series for f(x) = sin^2(x)?

[KMjuniormint5]

Homework Statement

Find the Fourier series of f(x) = sin^2(x)

Homework Equations

bn = because f(x) is even
ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏)
an = (1/(∏))*∫(f(x)*cos(x)) (from 0 to 2*∏)

The Attempt at a Solution

ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏) = ao = 1/2
an = (1/(∏))*∫(f(x)*cos(x)) (from 0 to 2*∏) = sin^3(x) from 0 to 2∏ and I keep resulting in zero

the answer is to the Fourier series I know is 1/2 - (cos(2x))/2 how to get the cos(2x)/2 part. Is there a trig identity I am missing?

 

[SammyS]

cos(2x) = 1 - 2 sin²(x)

 

[KMjuniormint5]

so in this case would you just plug cos(2x) into an to get it to be:

an = (1/(∏))*∫(sin^2(x)*(1-2*sin^2(x/2))dx

 

[vela]

Your integral is wrong. You should have
$$a_n = \frac{1}{\pi}\int_0^{2\pi} \sin^2 x \cos nx\,dx$$
The answer will depend on n, and you can't use the substitution u=sin x, like you seem to be doing, because the n is in the cosine.

 

[KMjuniormint5]

when I intergrate I get 1/4$\pi$*(cos(nx)*sin(2x)-2*x*cos(nx)) evaluated over 0 to 2$\pi$ which is still zero

 

[vela]

How'd you get that?

 

[KMjuniormint5]

an online integral calculator (numberempire.com) but i forgot the negative sign out front

 

[vela]

Well, it's not correct, but I also don't see how you got it equal to 0 when evaluated at 0 and 2pi.

You actually don't have to do a single integral for this problem, but it's probably worth evaluating the integrals to get the practice and to see that you do indeed get the correct answer. Try solving the identity SammyS noted for sin² x and using that in your integral. You'll also need to use the identity$$\cos a\cos b = \frac{\cos(a+b)+\cos(a-b)}{2}$$Take care that you don't divide by 0 as well.

 

[KMjuniormint5]

after going back and doing it the correct way (substitution) I get something of:
$_{}a_n$=$\frac{}{}2/\pi$*($\frac{}{}1/n$*sin^2(x)*sin(nx)+$\frac{}{}1/n^2$sin(2x)*cos(nx)+$\frac{}{}2/n^2$$\int cos(2x)*cos(nx)$dx

Using SammyS cos(2x)= 1 - 2sin^2(x) you can see that you get the original function of $_{}a_n$

so I get
($\frac{}{}2/\pi$+$\frac{}{}8/(\pi*n^2)$$\int sin^2(x)*cos(nx)$= $\frac{}{}4/(\pi*n^2)$$\int cos(nx)$

but when you evaluate the integral on the RHS wouldn't you get sin(nx) which evaluated from 0 to $\pi$ would be 0?

 

[vela]

When all you show is your result, all I can say is you're not doing something correctly. You seem to be making this a lot harder than it has to be, though.