User:Bosmon

A scratchpad for testing some maths markup, including the MITCH FORMULA, where $$f(t) = \sin(\alpha + \omega t)$$ and $$A = f(0), B = f(T), C = f(2T), D = f(3T)$$.

$${AD - BC\over BB - CA + CC - BD}$$

$$= {\sin(\alpha)\sin(\alpha + 3\omega T) - \sin(\alpha + \omega T)\sin(\alpha + 2 \omega T) \over(etc.)}$$

$$= {\cos(3\omega T) - \cancel{\cos(2\alpha + 3\omega T)} - \cos(\omega T) + \cancel{\cos(2\alpha + 3\omega T)} \over \cos(0) - \cancel{\cos(2\alpha + 2\omega T)} - \cos(2\omega T) + \cancel{\cos(2\alpha + 2\omega T)} + \cos(0) - \cancel{\cos(2\alpha + 4\omega T)} - \cos(2\omega T) + \cancel{\cos(2\alpha + 4\omega T)}}$$

$$ = {\cos(3\omega T) - \cos(\omega T)\over 2(1 - \cos(2\omega T))}$$

$$ = {-2\sin(2\omega T)\sin(\omega T)\over 4\sin^2(\omega T)} = {-\sin 2\omega T\over 2\sin \omega T} = {-2 \sin \omega T \cos \omega T\over 2\sin\omega T} = -\cos \omega T$$