complex angle?

$\cos(i\theta) + i\sin(i\theta)$ .

This equation gives a real value $e^{(-\theta)}$ . This shows that purely complex number can be equal to purely real number. But for this angle must be complex. Is this possible? Can angle be complex? I think in maths everything is possible. Initially no one knows about $x^2+1=0$ . So if angle can be imaginary then we will get our result to be real. So what does it signify?

Easy Math Editor

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Comments

Well, actually, we have the following definitions $\forall x \in \mathbb{C}$ $\begin{aligned} \cos x &= \frac{e^{ix}+e^{-ix}}{2} \\ \sin x &= \frac{e^{ix}-e^{-ix}}{2i}\end{aligned}$

The definition of trignometric ratios first started by using angles. Later, it cropped up in other places as well. So, the above analytic definition was accepted. Also, notice that for $x \in \left(0, \dfrac{\pi}{2} \right)$ , this definition coincides with the geometric one.