Discovering the Cosine Series of Sine: Insights and Calculations Explained

[tiagotorres]

I tried to find the cosine series of the function f(x) = \sin x, using the equation below:

S(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)
where: a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx

I found:

a_0 = \frac{4}{\pi}
a_n = \frac{2 }{\pi (1 - n^2)} (\cos(n \pi) + 1)

Therefore:

S(x) = \frac{2}{\pi} (1 - 2 \sum_{n=1}^{\infty} \frac{\cos(2nx)}{4n^2 - 1})

Making the graph of the first terms of the function above on my calculator, I noticed that this is actually | \sin x |, rather than just \sin x. Why does this happen? Is there a way of figuring out the sum not using a calculator?

Thanks

 

[mathman]

sinx is nonegative between 0 and pi, so sinx=|sinx| there.