consider pedagogically defining sine and cosine as the prettiest oscillating functions. figuring out how to construct the functions from this definition is left as an exercise for the student.
unlike the standard definitions which immediately invoke geometry, calculus, differential equations, complex numbers, or infinite series, often intimidating left-brain subjects, this definition asks for "pretty", a right-brain subject, an avenue of exploration that might be more appealing to some students.
or, "elegant".
perhaps the student will independently discover some intuitive form of the differential equation x'' = -x , or discover projecting circular motion onto a line, or discover "smooth", differentiable an infinite number of times, as a way of defining "pretty".
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