Euler's formula is, by all accounts, an impressive looking
beast. It brings together e, the complex plane, trigonometry
and real numbers. But let me tell you the deep truth about
working with it: there is nothing to solve! A problematic identity
has an x term in it, whose value one is looking for. Here, you
know the value of the angle, or that of one of the trig functions
and all falls ito place. This identity is about setting the affix point
on the unity circle. Or making the reference figurego up a size
with it's coefficient.
Below, some lovely applets allowing one to do just that.
https://www.intmath.com/complex-numbers/euler-formula-identity.php
https://www.intmath.com/complex-numbers/multiplying-dividing-complex-numbers-interactive.php
For those who lack clarity on how this all hangs together, a
Khan Academy video. Another spoiler: it has to do with the serie
which defines e, on which an alternates pick gives the cos and
sin functions.
https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-14/v/euler-s-formula-and-euler-s-identity
Most hardworking thing in the business!
Fun fact: the German-language Wikipedia article on de Moivre claims he
was a student of Isaac Newton.
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