Pi/Zero of cosine/Introduction/Section

From Wikiversity
< Pi
Jump to navigation Jump to search

This page is a small or not so small module that is part of modular mathematics. Its usefulness may only be apparent if one checks the what-links-here function to see in which course/larger page this module is used.

The number is the area and half of the circumference of a circle with radius . But, in order to build a precise definition for this number on this, we would have first to establish measure theory or the theory of the length of curves. Also, the trigonometric functions have an intuitive interpretation at the unit circle, but also this requires the concept of the arc length. An alternative approach is to define the functions sine and cosine by their power series, and then to define the number with the help of them, and establishing finally the relation with the circle.

The cosine function has, within the real interval , exactly one zero.

We consider the cosine series

For , we have . For , one can write

Hence, due to the intermediate value theorem, there exists at least one zero in the given interval.
To prove uniqueness, we consider the derivative of cosine, which is

due to fact. Hence, it is enough to show that sine is positive in the interval , because then cosine is strictly decreasing by fact in the interval and there is only one zero. Now, for , we have

A rational approximation of the number on a -pie.

Let denote the unique (according to fact) real zero of the cosine function in the interval . Then the number is defined by