What should be the correct range of acot function
06-06-2020, 06:42 AM
Post: #1
 teerasak Member Posts: 62 Joined: Jun 2019
What should be the correct range of acot function
As in the book, the range of acot(x) is 0 to pi

But when I check in HP prime by plotting acot(x) function, the range is -pi/2 to pi/2

Also, in wolfram alpha, the range is -pi/2 to pi/2

In TI-Nspire, the range is 0 to pi

What should be the correct one?
06-06-2020, 06:52 AM
Post: #2
 anyfoo Junior Member Posts: 22 Joined: May 2020
RE: What should be the correct range of acot function
Matlab also uses -pi/2 to pi/2. Both are valid ranges, and so would be e.g. 2pi to 3pi, although unusual. The cotangent, like the other trigonometric function, repeats with multiples of pi, so to define a function that is suitable for use as its inverse, you can take any interval of pi.
06-06-2020, 08:50 AM
Post: #3
 teerasak Member Posts: 62 Joined: Jun 2019
RE: What should be the correct range of acot function
Thank you, anyfoo. I understand that the function is defined as interval of pi. But when referring to a standard function, it must have an agreed range e.g asin(x) - the range is -pi/2 to pi/2, acos(x) is 0 to pi. Just see that acot has different range in calculation tools and the book.
06-06-2020, 01:20 PM
Post: #4
 pinkman Senior Member Posts: 387 Joined: Mar 2018
RE: What should be the correct range of acot function
(06-06-2020 08:50 AM)teerasak Wrote:  Thank you, anyfoo. I understand that the function is defined as interval of pi. But when referring to a standard function, it must have an agreed range e.g asin(x) - the range is -pi/2 to pi/2, acos(x) is 0 to pi. Just see that acot has different range in calculation tools and the book.

In “my” books, tangent is defined from ]-π/2, π/2[ to ℝ
And it’s reciprocal cotangent is defined from ℝ to ]-π/2, π/2[, but US Wikipedia says ]0, π[
06-06-2020, 02:34 PM
Post: #5
 ijabbott Senior Member Posts: 921 Joined: Jul 2015
RE: What should be the correct range of acot function
IMHO, since the "co" stands for "complementary", it seems to make more sense from a language point of view for atan(x) + acot(x) (in radians mode) to sum to pi/2 in the same way than asin(x) + acos(x) sum to pi/2.

— Ian Abbott
06-06-2020, 03:09 PM (This post was last modified: 06-07-2020 01:28 AM by Wes Loewer.)
Post: #6
 Wes Loewer Member Posts: 268 Joined: Jan 2014
RE: What should be the correct range of acot function
There is no consensus on the range of the acot(x) function.

Using -pi/2 < y < pi/2 comes from defining acot(x)=atan(1/x), consistent with asec(x)=acos(1/x) and acsc(x)=asin(1/x)

Using 0 < y < pi comes from defining acot(x)=supplement of atan(x), consistent with acos(x)=supplement of asin(x) and acsc(x)=supplement of asec(x).

Most US textbooks use (0,pi), but a few use (-pi/2,pi/2), and many simply don't mention acot at all. Each has its advantage: (0,pi) is continuous but (-pi/2,pi/2) preserves odd symmetry and can be calculated more precisely for negative values of x.

I bring up this issue in my Precalc class each year and have the students argue for their preference. Makes for a fun discussion.

The nice thing about standards is that you have so many to choose from. ~Andrew Tanenbaum

----
Edit: Sorry, I meant "complement" above, not "supplement." That's where the "co" in cosine, cotangent, and cosecant comes from.
06-06-2020, 05:39 PM
Post: #7
 teerasak Member Posts: 62 Joined: Jun 2019
RE: What should be the correct range of acot function
Thank you for clarification. That clarifies my doubt.
06-07-2020, 01:33 AM
Post: #8
 Wes Loewer Member Posts: 268 Joined: Jan 2014
RE: What should be the correct range of acot function
Quote: Using 0 < y < pi comes from defining acot(x)=supplement of atan(x)

Woops. I meant complement, not supplement. That's what cosine means: the complement sine.

(We don't have to mention this little error to my students. :-) )
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