# Deriving Euler’s Formula: e^ix = cos(x) + i sin(x) (VIDEO!)

Perhaps the most surprising and beautiful result in all of mathematics, Euler’s formula,
e^ix = cos(x) + i sin(x), turns the theory of trigonometry into a simple study of exponents. (And put in x = pi to get e^i*pi = -1).

In this video I descibe a way to get to this result at the beginning of a calculus course (once one knows some basic derivatives) rather then leaving it to the end of a course using Taylor series.

See another video on how I choose to derive the number e in a calculus course and yet another video as to why this is the same number e that arises in the theory of compound interest. (Full details, of course, appear in Volume 5 of the THINKING MATHEMATICS book.)

Euler’s Formula

• ## ON-LINE SHORT COURSES!

QUADRATICS, Permutations and Combinations, EXPLODING DOTS, and more!

Written for educators - and their students too! - this website, slowly growing, takes all the content Tanton has developed in his books, videos, and workshops, and organizes it into short, self-contained, and complete, curriculum units proving that mathematics, at all points of the school curriculum, can be joyous, fresh, innovative, rich, deep-thinking, and devoid of any rote doing! Let's teach generations of students to be self-reliant thinkers, willing to flail and to use their common sense to "nut their way" through challenges, to assess and judge results, and to adjust actions to find success. (Great life skills!)

CHECK OUT: www.gdaymath.com