Mathematical induction for differentiation Watch

andrewlee89
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Use mathematical induction to prove that for all positive integers n,

\frac{d^n}{dx^n}sinx = sin(x+\frac{1}{2}n\pi)

So to start off,

Let P(n) be \frac{d^n}{dx^n}sinx = sin(x+\frac{1}{2}n\pi)

Basis case, when n=1, \frac{d}{dx}(sinx) = cosx
sin(x+\frac{\pi}{2})=cosx

Suppose n=k,

\frac{d^k^+^1}{dx^k^+^1}(sinx) = \frac{d}{dx}(\frac{d^k}{dx^k}sin  x)

= \frac{d}{dx}(sin(x+\frac{1}{2}k\  pi)

= \frac{d}{dx}(sinxcos\frac{1}{2}k  \pi + cosxsin\frac{1}{2}k\pi)

Can anyone help me go from here? I'm nt sure how to clear off the k's in the angle right now...
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DFranklin
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= \frac{d}{dx}(sin(x+\frac{1}{2}k\  pi)

= \frac{d}{dx}(sinxcos\frac{1}{2}k  \pi + cosxsin\frac{1}{2}k\pi)

Can anyone help me go from here? I'm nt sure how to clear off the k's in the angle right now...
Instead of expanding \sin(x+\frac{k\pi}{2}), just use the chain rule.
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insparato
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Yeah and remember sinx and cosx are the same graph just a translation apart.
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andrewlee89
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(Original post by insparato)
Yeah and remember sinx and cosx are the same graph just a translation apart.
What does this mean?
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DFranklin
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(Original post by andrewlee89)
What does this mean?
cos x = sin(x+a) for a suitable choice of a.
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andrewlee89
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Oh.. Well that's true but this is proof by induction I need to get the whole equation into the k+1 format for proof
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DFranklin
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(Original post by andrewlee89)
Oh.. Well that's true but this is proof by induction I need to get the whole equation into the k+1 format for proof
Yes, we know it's proof by induction. How do you think you are going to go from \frac{k\pi}{2} to \frac{(k+1)\pi}{2}? Do you think it might possibly be by adding \frac{\pi}{2} at some point?
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andrewlee89
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(Original post by DFranklin)
Yes, we know it's proof by induction. How do you think you are going to go from \frac{k\pi}{2} to \frac{(k+1)\pi}{2}? Do you think it might possibly be by adding \frac{\pi}{2} at some point?
True :P Thx for the help. Will continue at it.
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lordcrusade9
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u got it yet?

cos x = sin (x+pi/2) or sin x = cos (x-pi/2) is the thing u need. it is actually a very short proof
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andrewlee89
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(Original post by lordcrusade9)
u got it yet?

cos x = sin (x+pi/2) or sin x = cos (x-pi/2) is the thing u need. it is actually a very short proof
Got it thx guys. U all rock. At various different times lol
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