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Find $f^{(2012)}(\pi/6)$ if $f(x)=\sin x$

Here's the hint from the question paper: You may use Maclaurin series of $\sin x$ and $\cos x$; the formula $\sin(a+b) = \sin a \cos b + \cos a \sin b$ may be useful.

It is quite complicated to me. Btw this is my math graded homework. I know how to find derivative when $x = 0$, but i can't proceed with this qn coz of the $\pi/6$. Any ideas?

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    I see, i see. Thanks for pointing that out. So it is not difficult at all.2012-10-28

2 Answers 2

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Notice $f(x) = \sin x$, $f'(x) = \cos x$, $f''(x) = - \sin x$, $f'''(x) = - \cos x$ and $f^{(IV)}(x) = \sin x$. How can you use this information to find the nth derivative of $f$ ? This is a hint.

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    @uohzxela Sometimes people, even sharp mathematicians, go for pretty messy stuff obviating simpler approaches. Read about the two trains puzzle and von Neumann's answer to it..and the astonishing way he obtained it here: http://mathworld.wolfram.com/TwoTrainsPuzzle.html2012-10-28
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$Let \space y=sin(x)$

$y'=cos(x)=sin(x+\pi/2)$

$y''=-sin(x)=sin(x+2(\pi/2))$

$y'''=-cos(x)=sin(x+3(\pi/2))$

$...$

$ y^{(n)}(x)=sin(x+n(\pi/2))$

You could now simplify this expression and your specific value: finding-the-values-of-cos-fracn-pi2-and-sin-fracn-pi2.