Why does an even signal times sine give an odd function in Fourier series?

[Peon666]

Hi all!

I wanted to have a little clarification about this line in relation to Fourier Series: (It's about a periodic and symmetrical signal)

"x(t) is a periodic signal. As cos nwt is an even function and sin nwt is an odd function. So, if x(t) is an even function of t, then x(t) cos nwt is also an even function and x(t) sn nwt is an odd function of t.

Similarly, if x(t) is an odd function of t, then x(t) cos nwt is an odd function of t and x(t) sin nwt is an even function of t."
- Linear Signal & Systems, B.P Lathi.

x(t) is both cos and sin functions multiplied right? And isn't it that an when an even function is multiplied with an odd function, we get an odd function? So how can be x(t) even? (as in the above excerpt)

Thanks.

 

[shazi]

"x(t) is both cos and sin functions multiplied right? "

The Fourier representation is the sum of cos and sin at different frequencies and amplitudes (instead of multiplying them) But to find the amplitudes we have to multiply the function being represented with cos and sin (along with more steps that can be found on wiki)