Find Real Value of sin(i) Without i

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Discussion Overview

The discussion revolves around finding the real value of sin(i), exploring methods such as series expansion and the use of imaginary exponentials. Participants are examining the nature of the result and whether there is a more straightforward expression for the real component.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests using series expansion to find the real value of sin(i) and questions if there is a nicer way to express the result.
  • Another participant references hyperbolic functions but does not elaborate on their relevance to the discussion.
  • A participant calculates sin(i) using imaginary exponentials, arriving at the expression ## \sin(i) = \frac{e^{-1} - e}{2i} ##, noting that it appears purely imaginary and expresses confusion about the real values from the series expansion.
  • Another participant confirms that sin(i) is purely imaginary and points out that the series expansion contains only terms with odd powers of i, which leads to imaginary results.

Areas of Agreement / Disagreement

Participants generally agree that sin(i) results in a purely imaginary value, but there is some contention regarding the interpretation of the series expansion and the presence of real values.

Contextual Notes

There is an unresolved question regarding the interpretation of the series expansion and its terms, particularly concerning the presence of real values in the context of sin(i).

TheCanadian
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I am calculating a value and want to find the real value of sin(i). I can use series expansion and only take the terms without i (correct?) but is there any nicer way to express the result of taking the real value of sin(i)?
 
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micromass said:

I used the imaginary exponentials and found ## \sin(i) = \frac{e^{-1} - e}{2i} ## but this seems purely imaginary...from the sum expansion of sin, it appeared that there were real values for the even power terms. Any advice you have for addressing what I am missing here since I'm looking for the real value would be great!
 
Indeed, ##\sin(i)## is purely imaginary. If you look at the series expansion of ##\sin(i)##, you'll see that only ##i^{\text{odd power}}## appear in this expansion. And as you know, ##i^{\text{odd power}} = \pm i##.
 

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