Show that a set $$ \{1, \sin x, \cos x, \sin 2 x, \cos 2 x, \ldots, \sin n x, \cos n x\} $$ are mutually orthogonal on $$ (-\pi, \pi) $$. Is this set orthogonal over $$ (0, \pi) $$

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Pierce Newton · Accepted Answer

The set $$ \{1, \sin x, \cos x, \sin 2x, \cos 2x, \ldots, \sin nx, \cos nx\} $$ is mutually orthogonal on the interval $$ (-\pi, \pi) $$. However, on the interval $$ (0, \pi) $$, the set is not orthogonal because the inner product between the constant function and certain sine functions does not vanish.

Show that a set \( \{1, \sin x, \cos x, \sin 2 x, \cos 2 x, \ldots, \sin n x, \cos n x\} \) are mutually orthogonal on \( (-\pi, \pi) \). Is this set orthogonal over \( (0, \pi) \)

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Show that a set \( \{1, \sin x, \cos x, \sin 2 x, \cos 2 x, \ldots, \sin n x, \cos n x\} \) are mutually orthogonal on \( (-\pi, \pi) \). Is this set orthogonal over \( (0, \pi) \)

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