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Hey, I'm wondering if anyone knows of a trig identity for arct(a/b) where a and b are rationals.

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Hey, I'm wondering if anyone knows of a trig identity for arct(a/b) where a and b are rationals.

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phyzguy

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http://functions.wolfram.com/ElementaryFunctions/ArcTan/27/02/

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Hi, perhaps you can tell us what you're doing. That would provide a context to work in.

In general, what's an arctan? If you have a line through the origin in the x-y plane, say it goes through some nonzero point (b,a). Then its slope is a/b, to be consistent with your notation.

The slope of the line is the tangent of the angle the line makes with the positive x-axis taken counter-clockwise. So the arctan of a/b is just the angle made by a line that passes through the origin and the point (b,a).

I don't know whether that's helpful or not, since I don't understand what you're trying to do. But it's one way to think about the arctan function, especially if you're already given the argument as a quotient (of two rationals or two reals, doesn't matter).

Another way to think about the arctan is to convert a complex number from rectangular to polar form. Given z = b + ai, to convert to polar form you end up taking the arctan of a/b (again reversing the usual use of a and b to conform with your notation).

And still a third way to think of the arctan is that it provides a handy gadget any time you need to continuously biject the entire real line onto a bounded open interval.

Perhaps one or more of these points of view will provide you with some insight into your problem.

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Well, arctan is defined for all real numbers. So no matter what a and b are, arctan(a/b) exists. Am I misunderstanding?

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Hurkyl

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(aside: a/b isn't defined for all real numbers....)Well, arctan is defined for all real numbers. So no matter what a and b are, arctan(a/b) exists. Am I misunderstanding?

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oo thanks Guffel!

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