Euler Formula

The formula which connects the complex exponential with trigonometric motion was discovered by Leonhard Euler around 1740, and it is called Euler's formula in his honor.

EULER'S FORMULA

\begin{equation*}e^{i\theta} = \cos \theta + i \sin \theta.\end{equation*}

It says that the complex number

e^{i\theta}

moves on the unit circle, with real part

\cos\theta

and imaginary part

\sin\theta

When

\theta=\pi

, Euler's formula gives the identity

\begin{equation*}e^{i\pi} + 1 = 0,\end{equation*}

which combines the constants

e

i

\pi

1

, and

0

in a single equation.

Applications

Trigonometry

Multiplying by

e^{i\theta}

rotates a point on the unit circle by the angle

\theta

Euler's formula gives a clean way to derive trigonometric identities. The key idea is that multiplication of complex exponentials turns addition of angles into multiplication:

\begin{equation*}e^{i(\theta+\varphi)}=e^{i\theta}e^{i\varphi}.\end{equation*}

Using Euler's formula on both sides, we get

\begin{equation*}\cos(\theta+\varphi)+i\sin(\theta+\varphi)=(\cos\theta+i\sin\theta)(\cos\varphi+i\sin\varphi).\end{equation*}

Expanding the product gives

\begin{equation*}\underbrace{(\cos\theta\cos\varphi-\sin\theta\sin\varphi)}_{\cos(\theta+\varphi)}+i\underbrace{(\sin\theta\cos\varphi+\cos\theta\sin\varphi)}_{\sin(\theta+\varphi)}.\end{equation*}

Now compare real and imaginary parts. The real parts give

\begin{equation*}\cos(\theta+\varphi)=\cos\theta\cos\varphi-\sin\theta\sin\varphi,\end{equation*}

and the imaginary parts give

\begin{equation*}\sin(\theta+\varphi)=\sin\theta\cos\varphi+\cos\theta\sin\varphi.\end{equation*}

Thus one complex equation contains two familiar trigonometric identities at once.

References

[Needham2023]Needham, Tristan. Visual Complex Analysis. 25th anniversary edition. Oxford University Press, 2023.