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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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pwsnafu

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But in order to do that you need to parametrize an arc...and how do you do that without sine and cosine?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.

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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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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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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