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Math Help - Prove these functions are orthogonal

  1. #1
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    Prove these functions are orthogonal

    Hi, can anyone help me solve this:

    2 functions f and g are orthogonal on [a,b] if:

    \int_a^b f(x)g(x)dx = 0

    Prove that f_{m}(x) = sin mx and f_{n}(x) = sin nx are orthogonal on the interval [-\pi,\pi] if m and n are integers such that m^2 n^2

    Thanks.
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  2. #2
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    Quote Originally Posted by coldfire View Post
    Hi, can anyone help me solve this:

    2 functions f and g are orthogonal on [a,b] if:

    \int_a^b f(x)g(x)dx = 0

    Prove that f_{m}(x) = sin mx and f_{n}(x) = sin nx are orthogonal on the interval [-\pi,\pi] if m and n are integers such that m^2 n^2

    Thanks.
    Please post what you've done and state where you get stuck.

    Note that \sin (nx) \sin (mx) = \frac{1}{2} \left(\cos ([n - m]x) - \cos ([n + m]x) \right).
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by coldfire View Post
    Hi, can anyone help me solve this:

    2 functions f and g are orthogonal on [a,b] if:

    \int_a^b f(x)g(x)dx = 0

    Prove that f_{m}(x) = sin mx and f_{n}(x) = sin nx are orthogonal on the interval [-\pi,\pi] if m and n are integers such that m^2 n^2

    Thanks.
    Using the identitiy that

    \sin(u) \sin(v)=\frac{1}{2}(\cos(u-v)-\cos(u+v))

    So in your case you get

    \sin(mx) \sin(nx)=\frac{1}{2}(\cos((m-n)x)-\cos((m+n)x))

    It should all be down hill from here.

    EDIT: haha too slow
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