Understanding the Differences in Arctan Values: A Scientific Perspective

[wirefree]

G'day!

I am in need of a nudge. My basic math understanding is being called to question by the arctan function.

Two values of arctan(-3) result:
a) In my textbook: 108.43 degrees
b) In my calculator: -71.56 degrees

It's perplexing and my superior has not put in a kind word either.

Would greatly appreciate encouragement.

Best Regards,
sobers

 

[andrewkirk]

Draw radial lines from the origin at those two angles on the number plane and you'll see that they are 180 degrees apart. The tangent function is periodic, repeating itself every 180 degrees, so both those angles have the same tangent. Which angle you want to use will depend on the context. Usually in the context of a specific problem, only one of the possible angles makes sense.

 

[Erland]

Your calculator is correct and your textbook is wrong. By the standard definition of arctan, its range is the interval ]-π/2, π/2[. Converting to degrees (interpreting the original values as given in radians), this means that arctan(x) lies between -90 degrees and 90 degrees. It is true that both 108.43 degrees and -71.56 degrees both have the tangens value -3 (within error limits), but only the second one equals arctan(-3).

So, your textbook seems to be quite poor.

This is actually to stretch things a bit, since it is very uncommon to use arctan (and arcsin and arccos) and degrees in the same context; radians are always implicitly assumed.

 

[wirefree]

Your calculator is correct and your textbook is wrong.

I am indebted for your honest assessment, Sir. The quality of the textbook has been a cause of concern for me on other fronts as well, which are:

a) Presence of typogaphical errors, not of the kind this post relates to, is quite consistent, which makes me wonder about 'developing countries' versions of American textbooks. Surely, large publishers, that rightly cut costs on printing papers & colors to allow for heavy discounts that these books come with, won't be cutting corners on actual proof-reading of the material. That can't surely be. I mean, once the book is ready for publishing, it's ready for publishing anywhere.

That only leaves the possibility that typographical errors exist in hard-bound, full-color versions too. And that does make this a poor textbook.

b) Any author that proceeds through a proof by simply stating 'Next, taking the curl of..." doesn't offer much by way of intuitive understanding. Admittedly, taking the curl of anything relates to an understanding of its rotationality, but in the context of, say, proceeding from one of Maxwell's equation to a general wave equation, a comment in english on the general idea, direction and motivation is surely the hallmark of a considerate teacher.

But that's a topic for another post.wirefree

 

[FactChecker]

There is a "standard" definition of arctan, but you often need to use the other values in a particular problem. If your textbook was using this in a larger problem, that may explain why they use that value. In any case, you should be aware that you will often need to use the other possible values of arctan for things to make sense.

 

[Erland]

There is a "standard" definition of arctan, but you often need to use the other values in a particular problem. If your textbook was using this in a larger problem, that may explain why they use that value. In any case, you should be aware that you will often need to use the other possible values of arctan for things to make sense.

Then it shouldn't be called arctan. arctan is the inverse of the branch of the tangent function restricted to ]-π/2, π/2[ (or ]-90°,90°[ ). If we need anther solution of the equation tan(x) = b than x = arctan(b), then we should write x = arctan(b) + nπ (or x = arctan(b) + n × 180°) for some suitable integer n.

 

[FactChecker]

Then it shouldn't be called arctan. arctan is the inverse of the branch of the tangent function restricted to ]-π/2, π/2[ (or ]-90°,90°[ ). If we need anther solution of the equation tan(x) = b than x = arctan(b), then we should write x = arctan(b) + nπ (or x = arctan(b) + n × 180°) for some suitable integer n.

Computer languages usually (always?) define the range of atan and atan2 that way, but I am not sure that mathematics specifies that arctan indicates an official principle branch.
CORRECTION: In most computer languages, atan() returns a value in [-π/2, π/2] and atan2() returns a value in [-π, π] according to the sign of its inputs.
Apparently there are some people who use Arctan and Tan^-1 to indicate that principle branch that you are specifying. For them, arctan is not necessarily the principle branch.
I am not sure how standardized the notation is within mathematics.

That being said, I have to admit that your approach seems wise. But I would hesitate to condemn someone else without knowing if there had been anything he said about the arctan function before this use of it.

 

[Erland]

I have never seen arctan being defined in any other way than this "principal branch".

And I didn't condemn anyone except possibly some textbook author... And this seems reasonable by other reasons, since OP told us that this book is poor in other ways too...

 

[FactChecker]

I have never seen arctan being defined in any other way than this "principal branch".

Ok. I'll buy that. I do think it is wise to define it as you say. In any case, a person should be alert that he may need to add nπ sometimes to make calculations work out correctly. That is true no matter how arctan is defined.