Euler Formulas

Formulas connecting the exponential and trigonometric functions:

$$e^{iz}=\cos z+i\sin z,$$

$$\cos z=\frac{e^{iz}+e^{-iz}}{2},\quad\sin z=\frac{e^{iz}-e^{-iz}}{2i}.$$

These hold for all values of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become

$$\cos x=\frac{e^{ix}+e^{-ix}}{2},\quad\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$

These formulas were published by L. Euler in [1].

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