Complex Exponentials, trigonometric functions Part 4 - What is pi ?
Description
We will now take an even closer look into Euler’s formula this time. More precisely we will investigate whether there exists a real number such that the imaginary unit times this real number plugged into the complex exponential function will give the imaginary unit as a result. In fact, as it will turn out, this is equivalent to finding roots of the cosine function. Having identified the cosine function as a continuous function we the apply the intermediate value theorem to obtain existence of such a zero. The existence of such a zero alone is sufficient to obtain periodicity and symmetry properties for both sin and cos. The only zero of cos in the interval from 0 to 2 will be called pi/2, which gives a stand alone definition of the precise value of pi.
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