complex analysis - Is it possible to calculate trigonometric functions with the imaginary exponential

By Eulers identity $e^{i \theta}=\cos(\theta)+i \sin(\theta)$ sine and cosine can be written in exponential form as

$\sin(\theta)=\frac {e^{i \theta}-e^{-i \theta}}{2i}$ and $\cos(\theta)=\frac {e^{i \theta}+e^{-i \theta}}{2}$.

Could you calculate specific values of the trigonometric function with these formulae? My guess is that the complex exponential can only be calculated using Euler's identity so you have to know the values of sine / cosine to begin with. Is there any method to calculate the value of sine / cosine using the identity above? Is there any reason why it isn't (is?) possible?

Answer

You can evaluate it at imaginary values:

$$\sin(i)=\frac{e^{-1}-e}{2i}$$

$$\cos(i)=\frac{e^{-1}+e}2$$

But not much else.

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Good for deriving some formulas though:

$$\cos^2(x)=\left[\frac{e^{ix}+e^{-ix}}2\right]^2=\frac{e^{2ix}+2+e^{-2ix}}4=\frac{\cos(2x)+1}2$$