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.

Difficulty level: Moderate task
Proving or derivation task
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