Is every sine and cosine orthogonal to every other?

Question

I've been learning about Fourier series, and haven't found an explicit statement of this requirement for constructing any arbitrary function using just sines and cosines, so I'm asking here. Is it true that $\sin{ax},\sin{bx},\cos{cx},\cos{dx}$ are all orthogonal to each other for all distinct real $a,b,c,d$? Symbolically: $$\int \sin{ax} \sin{bx} = \int \sin{ax} \cos{cx} = \int \sin{ax} \cos{dx} = \int \sin{bx} \cos{cx}\; ...=0 $$ Is this easy to show?

Answer 1

I'll assume the range of integration is $[-\pi,\pi]$ and that $a,b$ are nonzero integers.

$\int_{-\pi}^{\pi} \sin(a x)\cos(bx)\,dx=0$ because the integrand is odd and integrable.

If $a\neq b$, then $\int_{-\pi}^{\pi} \sin(a x)\sin(bx)\,dx=\frac{1}{2}\int_{-\pi}^{\pi} -\cos((a+b)x)+\cos((a-b)x)\,dx=0$. Similarly for the double cosine case: $\int_{-\pi}^{\pi} \cos(a x)\cos(bx)\,dx=\frac{1}{2}\int_{-\pi}^{\pi} \cos((a+b)x)+\cos((a-b)x)\,dx=0$.