Scalar product of functions

Task number: 2692

Without calculating the integral show that for any $$a,b,r\in \mathbb R$$, $$a,b\ne 0, r>0$$ have functions $$f_a(x)=\sin(ax)$$ and $$g_b(x)=\cos(bx)$$ zero scalar product, i.e. they are orthogonal.

The product is given as: $$\langle f_a|g_b \rangle=\int_{-r}^r f_a(x)g_b(x)dx$$.

• Resolution

The function $$\sin(ax)$$ is odd, while $$\cos(bx)$$ is even. Their product $$f_a(x)g_b(x)$$ is an odd function. For any odd function integrated on a symmetric interval $$(-r,r)$$ holds that the integral is zero.