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

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?

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## Comments

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TopNewestWell, actually, we have the following definitions \(\forall x \in \mathbb{C}\) \[\begin{align} \cos x &= \frac{e^{ix}+e^{-ix}}{2} \\ \sin x &= \frac{e^{ix}-e^{-ix}}{2i}\end{align}\]

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.

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Thank you, @Deeparaj Bhat . But I still want to know what does it mean?. I mean to say complex number doesn't exist. Yet If we add complex angle to complex number then it results to real 'an existing one'

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The notion of angle is not used in the definition given above. As such, it doesn't make any reference to diagrams.

I don't understand what you mean by 'complex number doesn't exist'.

Also, real numbers are as imaginary as complex numbers. All these number systems are constructed by us. In fact, manier times complex numbers are used in real life situations.

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No. We do make new things. How do you have decimals in nature? We created it to simplify stuff.

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Decimals in nature. it has also relation with nature. Observation is medium to know about nature. Initially our observation was limited to only different things. So we discovered numbers 1,2,3,. Then we have observed in the particular thing.We thought what is there in the thing? Similarly we thought what is after '1'. For any particular thing, there is always something inside it. In this way decimal may be discovered.

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Coming back to your question, I think it has been answered.

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