Euler's Formula and Complex Trig Functions

Math Trig

Euler’s Formula

Euler’s formula states:

$$e^{i\theta }=i\sin \theta +\cos \theta$$

Proof

$$\begin{gathered} \frac{d}{d\theta }\frac{i\sin \theta +\cos \theta }{e^{i\theta }} \\ =e^{-i\theta }(i\sin \theta +\cos \theta ) \\ =(e^{-i\theta })(i\sin \theta +\cos \theta )\prime +(e^{-i\theta })\prime (i\sin \theta +\cos \theta ) \\ =(e^{-i\theta })(i\cos \theta -\sin \theta )-i(e^{-i\theta })(i\sin \theta +\cos \theta ) \\ =(e^{-i\theta })(i\cos \theta -\sin \theta )-(e^{-i\theta })(i\cos \theta -\sin \theta ) \\ =0 \end{gathered}$$

Therefore $\frac{i\sin \theta +\cos \theta }{e^{i\theta }}$ is a constant. Plug in $\theta =0$, to get $\frac{i\sin \theta +\cos \theta }{e^{i\theta }}=1$. Multiply both sides by $e^{i\theta }$ to get

$$e^{i\theta }=i\sin \theta +\cos \theta$$

Euler’s Identity

Plug $\theta =\pi$ into Euler’s Formula

$$\begin{gathered} e^{i\pi }=i\sin \pi +\cos \pi \\ {e}^{i\pi }=-1 \end{gathered}$$

Trig Functions Redefined

Sine:

$$\begin{gathered} e^{i\theta }=i\sin \theta +\cos \theta \\ -e^{-i\theta }=-i\sin -\theta -\cos -\theta \\ -e^{-i\theta }=i\sin \theta -\cos \theta \\ {e}^{i\theta }-e^{-i\theta }=2i\sin \theta \\ \sin \theta =\frac{e^{i\theta }-e^{-i\theta }}{2i} \end{gathered}$$

Cosine:

$$\begin{gathered} e^{i\theta }=i\sin \theta +\cos \theta \\ {e}^{-i\theta }=i\sin -\theta +\cos -\theta \\ {e}^{-i\theta }=-i\sin \theta +\cos \theta \\ {e}^{i\theta }+e^{-i\theta }=2\cos \theta \\ \cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2} \end{gathered}$$