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 sinx and cosx 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 sinx and cosx in the usual way, because it was a calculus class.

Question

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 sinx and cosx functions, which are actually defined using the perimeter of a circle, aren&#039;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&#039;t know but most probably they defined sinx and cosx in the usual way, because it was a calculus class.

Holden Rosario · Accepted Answer

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  =      ∑          n      =      0              ∞                          (        −        1                  )          n                          x                      2            n            +            1                                      (        2        n        +        1        )        !            and  cos  ⁡  x  =      ∑          n      =      0              ∞                          (        −        1                  )          n                          x                      2            n                                      (        2        n        )        !            which are just their Taylor Series representations.

Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using

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