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

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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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. – Deeparaj Bhat · 4 months ago

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@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' – Akash Shukla · 4 months ago

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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. – Deeparaj Bhat · 4 months ago

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– Akash Shukla · 4 months ago

for existence, I mean to say that we cannot say we have 'i' number of things. Also we cannot get 'i' grams of anything. And in practical life,I think, we always neglect the complex terms. I think we donot make new thing, Only just we observe and do accordingly and it becomes a new thing , then we say that we have constructed this !.Log in to reply

No. We do make new things. How do you have decimals in nature? We created it to simplify stuff. – Deeparaj Bhat · 4 months ago

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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. – Akash Shukla · 4 months ago

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Coming back to your question, I think it has been answered. – Deeparaj Bhat · 4 months ago

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– Akash Shukla · 4 months ago

The last line of question is the main question.Log in to reply

– Deeparaj Bhat · 4 months ago

Like I've already said before, in the modern definition, angles aren't used. It has no physical significance. Also, \(\sin\) is imaginary for imaginary values and \(\cos\) is real for imaginary values.Log in to reply

– Akash Shukla · 4 months ago

How can it has no significance? Everything has its own significance. Ok, you have explained it.Log in to reply

– Deeparaj Bhat · 4 months ago

If you say so.Log in to reply