This is true for any real number x.
Such a close connection between trigonometric functions, the mathematical constant "e", and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view.
In fact, Euler's formula has other surprises in store. If you substitute the value for x in Euler's formula, then, since cos = -1 and sin = 0, you get the identity
Rewriting this as
you obtain a simple equation that connects the five most common constants
of mathematics: e, , i , 0 , and 1.
Not the least surprising aspect of the last equation is that the result of raising an irrational number to a power that is an irrational imaginary number can turn out be a natural number. Indeed, raising an imaginary number to an imaginary power can also give a real-number answer. Setting x =/ 2 in the equation at the top of the page, and noting that cos / 2 = 0 and sin / 2 =1 , you get
and, if you raise both sides of this identity to the power i, you obtain
(since = -1)
Thus, using a calculator to compute the value of , you find that
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