Prove these functions are orthogonal

**coldfire** · October 20th 2009, 05:48 PM

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.

**mr fantastic** · October 20th 2009, 05:58 PM

Originally Posted by coldfire

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 $\sin (nx) \sin (mx) = \frac{1}{2} \left(\cos ([n - m]x) - \cos ([n + m]x) \right)$ .

**TheEmptySet** · October 20th 2009, 05:59 PM

Originally Posted by coldfire

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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