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Alex B.
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I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Alex B.
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