Euler’s Formula

A-11.3 Euler’s Formula

$⟨$ Definition A-11.3 $⟩$ Euler’s Formula

$e^{𝑖𝜃} = \cos 𝜃 + i \cdot \sin 𝜃$

Figure 7: The figure illustrates Euler’s formula on the unit circle in the complex plane (Source)

The value of $e^{𝑖𝜃}$ is represented as a coordinate on a circle in the complex plane in Figure 3, where the $x$ -axis encodes the value’s real number part and the $y$ -axis encodes the value’s imaginary number part. Note that as $𝜃$ increases, the imaginary part ( $\sin 𝜃$ ) oscillates between $1$ and $- 1$ (reaching $i$ and $- i$ on the imaginary axis), and the real part ( $\cos 𝜃$ ) oscillates between $1$ and $- 1$ , with period $2 π$ .

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