Euler's formula

Euler's formula is based on Maclaurin's expansion.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

x is 0,
so f(x)=(sin x,cos x,-sin x,-cos x,・・・)=(0,1,0,-1,・・・)=sin x
and f(x)=(cos x,-sin x,-cos x,sin x,・・・)=(1,0,-1,0,・・・)=cos x

Euler's identity is well known as the beauty of mathematics.