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It is a problem given next to the definition of Periodic function

The textbook problem is to prove $\sin(\pi/6)=1/2$

It is Already proved that $\sin(\pi/2)=1$ and $\sin$ addition formula...

Using that I easily proved $\sin(\pi/6)=1/2$,

But my doubt is, do the author wants the reader solve it using some sort of periodicity... is there any other method to prove it using the periodicity definition and all the details given above including the definition of sin using exponential function.

I tried and failed, so I just wanted to know whether we can or not Kindly help... TIA

Praveen Kumaran P
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  • @CameronWilliams Yes... but the definition of sine here is rigorous, which uses exp function and infinite series kinda things... so we need to proceed accordingly – Praveen Kumaran P Sep 27 '23 at 04:57
  • @CameronWilliams changed the title... thank you..:) – Praveen Kumaran P Sep 27 '23 at 05:00
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    I don't see any way periodicity can directly help here. Maybe the problem is in the section just because it involves some of the same steps and ideas used to prove that $\sin$ is periodic. – aschepler Sep 27 '23 at 05:09
  • @aschepler yes... sorry after this exercise, there is a exercise asking that "Under what conditions $\sin w = \sin z$?". So I think the author doesn't really want the reader to use the periodicity to solve it... because the next question just only talks about the periodicity of Sine – Praveen Kumaran P Sep 27 '23 at 05:18
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    If you already proved sine addition formula it isn't too far stretch to prove triple angle formula for sine, so $\sin(\pi/6)$ is either $-1$ or $\frac12$. But by definition $\pi$ being the first positive zero of sine and $\sin x>0$ for small $x>0$ we reject the $-1$. – user10354138 Sep 27 '23 at 05:34
  • @user10354138 yes I proved it using the same argument... – Praveen Kumaran P Sep 27 '23 at 06:09
  • I'm not quite sure exactly what you want, but this question https://math.stackexchange.com/questions/1863239/if-f-is-a-smooth-real-valued-function-on-real-line-such-that-f0-1-and-f tells us that if $f$ is a differentiable real perodic function, and if $f'$ is a shifted version of $f$, then $f(x)=A\sin(x+\phi)$ for some constants $A$ and $\phi$. Of course, the proof in that post uses properties of trigonometric functions. – Ningxin Sep 27 '23 at 09:35

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