What should be the correct range of acot function

06062020, 06:42 AM
Post: #1




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 TINspire, the range is 0 to pi What should be the correct one? 

06062020, 06:52 AM
Post: #2




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.


06062020, 08:50 AM
Post: #3




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.


06062020, 01:20 PM
Post: #4




RE: What should be the correct range of acot function
(06062020 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[ 

06062020, 02:34 PM
Post: #5




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 

06062020, 03:09 PM
(This post was last modified: 06072020 01:28 AM by Wes Loewer.)
Post: #6




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. 

06062020, 05:39 PM
Post: #7




RE: What should be the correct range of acot function
Thank you for clarification. That clarifies my doubt.


06072020, 01:33 AM
Post: #8




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. :) ) 

03212021, 06:21 PM
(This post was last modified: 03212021 06:25 PM by Albert Chan.)
Post: #9




RE: What should be the correct range of acot function
(06062020 03:09 PM)Wes Loewer Wrote: There is no consensus on the range of the acot(x) function. All these programs have range of acot(x) = [pi/2, pi/2]: 1. Mathematica 2. Sympy 3. Maxima 4. Python mpmath 5. XCas 6. HP Prime, both Home + Cas Is there any exceptions you know ? (e.g. acot(1) return 3/4*pi, instead of 1/4*pi ?) 

03252021, 06:58 AM
Post: #10




RE: What should be the correct range of acot function
Excel (Version 16.46 for Mac)
ARCCOT(1) =2,35619449 Documentation states that the results of ARCCOT are in the range between 0 and Pi Felix 

03252021, 07:14 PM
Post: #11




RE: What should be the correct range of acot function
(03212021 06:21 PM)Albert Chan Wrote: All these programs have range of acot(x) = [pi/2, pi/2]: 7. Google Sheets (03212021 06:21 PM)Albert Chan Wrote: Is there any exceptions you know ? (e.g. acot(1) return 3/4*pi, instead of 1/4*pi ?) . Excel, OpenOffice/LibreOffice, Quattro Pro . Desmos . GeoGebra (Well, sort of. It converts acot(x) to "pi/2atan(x)") . TINspire Not a program, but https://en.wikipedia.org/wiki/Inverse_tr...pal_values. 

03252021, 10:49 PM
(This post was last modified: 03262021 06:06 AM by Felix Gross.)
Post: #12




RE: What should be the correct range of acot function
Here some more data points
pi /2 to pi/2: 8. Sage: acot(1) = 1/4 PI 9. Matlab (as already mentioned in the thread) 10. NIST Handbook of Mathematical Functions, 2010, p 112, Fig. 4.15.4 0 to pi: Grapher (Mac program) Edit: Added NIST reference 

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