# Check the orthogonality of two functions

This is a part of two Passband Signals. My question involves inner product. Is
$$\left < \cos(2\pi ft + \phi) , \sin(2\pi ft + \phi) \right>$$ where $\phi \in [-\pi, \pi)$ equal to zero, i.e. that functions are orthogonal? Thank you in advance.

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@Kaster: Thank you for editing. –  karl71 Apr 10 '13 at 8:29
Which inner product are you using? –  Dennis Gulko Apr 10 '13 at 8:35
@DennisGulko: imagine <u,v>, so the inner product will be the result of integrating from -infinite to infinite the product of (u)(v*) –  karl71 Apr 10 '13 at 8:41
You have to integrate over 1 period, not from $-\infty$ to $\infty$. –  Matt L. Apr 10 '13 at 8:42
+1 for posting interesting question –  dato datuashvili Apr 10 '13 at 8:43

Yes, they are orthogonal. Consider the interval $[0,T]$ with $T=1/f$ and integrate, then you'll see it. In this case the inner product is defined by
$$\int_{0}^T \sin\left(\frac{2\pi t}{T}+\phi\right) \cos\left(\frac{2\pi t}{T}+\phi\right)dt$$
The value of $\phi$ doesn't matter as long as it shifts both function values equally. You always integrate over one period $T$, and it doesn't matter where you start. –  Matt L. Apr 10 '13 at 8:40