Check the orthogonality of two functions

Question

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.

Answer 1

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

$$\int_{0}^T \sin\left(\frac{2\pi t}{T}+\phi\right) \cos\left(\frac{2\pi t}{T}+\phi\right)dt$$