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Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using arc lenghts. Even though they seemed to be on the right track, there is a problem:

They are using $\sin x$ and $\cos x$ functions, which are actually defined using the perimeter of a circle, aren't they? So, in my opinion, that was not a real proof since they were using what they want to show.

Or am I missing something? Was that a real proof?

Edit: I don't know but most probably they defined $\sin x$ and $\cos x$ in the usual way, because it was a calculus class.

pjs36
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ThePortakal
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  • $\sin$ and $\cos $ can be defined in loads of ways. – A little lime Apr 17 '14 at 13:52
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    You can use perimeter $=2 \pi r$ to define $\pi$. In that case, you have to prove $\sin$ and $\cos$ have the properties you want. Or you can define $\sin$ and $\cos$ from (for example) their series, in which case you want this proof. There are similar issues with $\exp$ and $\ln$. You have to start somewhere, then prove all the other definitions follow from the one you chose. – Ross Millikan Apr 17 '14 at 14:12

1 Answers1

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$\sin x$ and $\cos x$ can be defined using the unit circle, but there are alternate definitions. For example, $\sin x$ and $\cos x$ can be defined as:

$$\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}$$

and

$$\cos x=\sum_{n=0}^{\infty}\dfrac{(-1)^nx^{2n}}{(2n)!}$$

which are just their Taylor Series representations.

Sujaan Kunalan
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