Motivation of sin and cos functions

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Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
 

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pwsnafu
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How are you defining the argument of a complex number without knowledge of sine and cosine?
 
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Fredrik
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In principle, the argument could be defined using the definition of "radian", but you'd have to use an integral to define the length of a curve segment. I think it would be difficult to use this approach to show that the arguments of the factors add up.
 
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pwsnafu
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In principle, the argument could be defined using the definition of "radian", but you'd have to use an integral to define the length of a curve segment.
But in order to do that you need to parametrize an arc...and how do you do that without sine and cosine?
Edit: Ah, wait I guess if you used y in terms of x, it would work. Never mind.
 
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mathwonk
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I don't understand your question as asked. I.e. how could one know the Taylor expansions before knowing the functions? To me the sin and cos functions are merely inverses of the arclength function. I.e. consider the unit circle, and define a function of y to be the arclength of the circle measured from the x axis to the point of the circle at height y. This is a natural function. The inverse of this function is the sin function. I.e. given the arclength of a portion of the circle reaching from the x axis to some point above it, (but remaining in the first quadrant), the y coordinate is the sin of that angle measured in radians. The cosine is similar.


Having read it again, your question does not seem to be how to motivate sina nd cos but how to prove the angle of a complex product is the sum of the angles of the factors. I.e. how to relate x and y coordinates of points in the plane to their polar coordinates. But that is essentially the meaning of sin and cos.
 
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Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
Yes. The book "Visual Complex Analysis" has a very nice development in the first 15 pages that only establishes Euler's formula after the basic properties of the exponential, it's Tayler series, and the Taylor series of it's real and imaginary parts have been established. The sin and cos are almost afterthoughts.
 
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