Complex Number Trigonometry

You can represent the trigonometric functions as complex numbers

cosθ=eiθ+eiθ2\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} sinθ=eiθeiθ2i\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} tanθ=sinθcosθ=1ieiθeiθeiθ+eiθ=ieiθeiθeiθ+eiθ\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{1}{i}\cdot\frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}} = -i\cdot\frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}}