Orthonomality Proof

$\dfrac{1}{T_0} \displaystyle\int^{t_0+T_0}_{t_0} \cos((k-l)\omega_0 t+j\sin(k-l)\omega_0 t)\ dt = 0 , k\neq l$

Hello everyone!

I know it sounds stupid, but I can't understand how does that expression becomes equal to zero when k is not equal to l.

Any help would be appreciated!

$\dfrac{1}{T_0} \displaystyle\int^{t_0+T_0}_{t_0} \cos((k-l)\omega_0 t+j\sin(k-l)\omega_0 t)\ dt = 0 , k\neq l$

Hello everyone!

I know it sounds stupid, but I can't understand how does that expression becomes equal to zero when k is not equal to l.

Any help would be appreciated!

What's $T_0$ ?

$\frac{2\pi}{\omega_0}$ by any chance?

If you perform the integration, you'll get a sin - sin, and a cos - cos.

Use the sum product rules/transformations for differences of sines and differences of cosines. One of the terms in each product will evaluate to zero.

is the sine meant to be inside the argument of the cosine?

Yes. $T_0 = \frac{2\pi}{\omega_0}$. Thank you. I'll work on it.



No, sorry. I've edited the original post now.