I was reading how to define the sine and cosine functions via their Taylor series. I think the hardest part is showing their periodicity. Some questions and answers to this include this and this. I understood how to show it. The key part is to define the number $\pi/2$ by the smallest positive number whose cosine is zero.
However, I could not find any proof that this definition of $\pi$ indeed coincides the geometric definition of $\pi$ in our mind, i.e., the ratio of a circle's circumference to its diameter. How can I prove this? If this question is a duplicate, I would appreciate it if you let me know. Thank you.
