Trig Identity for Arctan(a/b) with Rational Values?

[iceblits]

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

 

[phyzguy]

The wolfram functions site is a good resource:

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

 

[iceblits]

Thanks for the reply. Although I can't find arctan(a/b), this is still very helpful for what I'm doing.

 

[SteveL27]

Thanks for the reply. Although I can't find arctan(a/b), this is still very helpful for what I'm doing.

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.

 

[iceblits]

I wanted to show that arctan(a/b) may be written in the form arctan(a*m)+arctan(b*n) (or something like that) as part of a proof I am writing for a project. The entire explanation is long winded and it would take some time to explain but basically if I know that (in my project) all arctan(a) and arctan(b) and any linear combination of those exist but I have yet to show if all arctan(a/b) exist or not which is why I was hoping for a trig identity that would neatly answer the question

 

[SteveL27]

I wanted to show that arctan(a/b) may be written in the form arctan(a*m)+arctan(b*n) (or something like that) as part of a proof I am writing for a project. The entire explanation is long winded and it would take some time to explain but basically if I know that (in my project) all arctan(a) and arctan(b) and any linear combination of those exist but I have yet to show if all arctan(a/b) exist or not which is why I was hoping for a trig identity that would neatly answer the question

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

 

[iceblits]

But i want it to exist in the constraint of my problem. In my problem, all arctan(a) and arctan(b) exist but I don't know if all arctan(a/b) exist in my problem. For example, I know that arctan(I) where I is an irrational number does not exist in my problem

 

[Hurkyl]

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

(aside: a/b isn't defined for all real numbers...)

 

[Guffel]

Check out this post.

 

[iceblits]

oo thanks Guffel!