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I need to find the values of $a$ at which $$\lim_{h \to 0} \frac{\sin(a+h)-\sin(a)}{h} = 0.$$ I know that this means that we are looking for the values of $a$ at which $\dfrac{d}{dx}\sin x=0$, or $\cos x=0$. I also know from calculus 1 that a should equal $\dots,\dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dots$

However, I can't figure out how I can show this using "real-analysis" level definitions and theorems. Do I need to use the definition of limits to show that $\frac{\sin(a+h)-\sin(a)}{h} \to 0$? How do I use that to find the appropriate values of $a$?

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    It might help us if you told us your definition of sine.2012-11-11
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    My definition of sin? I'm not sure if I understand the request.2012-11-11
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    How are you defining the sine function? That will impact how the proof is given.2012-11-11
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    I'm not sure if this answers your question at all, but sinx is defined for all real numbers.2012-11-11
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    You're interested in a "real analysis" level proof. That means you have to start with with a rigorous definition of the sine function. So again I ask, what _is_ the sine function? Are you defining it as a power series? In terms of the complex exponential? As the solution of a differential equation?2012-11-11
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    Oh wow. I'm really sorry for my stupidity, but I don't think that we've gotten quite to that level yet in my analysis class.2012-11-11
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    In that case, I would say that the traditional proof that the cosine is the derivative of sine is rigorous enough until you have a formal definition of the sine function.2012-11-11
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    Would it be okay although we have not yet formally gone over the power rule in our course yet?2012-11-11
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    I'm not sure what you mean by that. The power rule doesn't play a role in proving that the derivative of sine is cosine. I assume you're just using the geometric definition of "opposite over hypotenuse", so have you seen the geometric proof based on the squeeze theorem?2012-11-11
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    Oh right, sorry. The power rule wouldn't matter for the derivative of sin. The geometric proof based on the squeeze theorem doesn't sound familiar, but I do know the squeeze theorem.2012-11-11
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    Hm. Have you seen a proof that the derivative of the sine is cosine at all? Normally such a proof is given in calculus before real analysis.2012-11-11
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    Ahhh yes. I do remember that. I guess I could use that proof to show that the derivative of sinx is cosx and then show the values of a at which cosx=0? Thank you so much for your guidance!2012-11-11

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