arctan(x) a multivalued function?

The single argument arctangent returns the Principal Value of the function, which ranges from -pi/2 to +pi/2.

There is such a thing as a two-argument arctangent function. This is the complete form that corrects for the quadrant of the angle.

They are related like this.

Let arctan(y/x) = the principal value of the arctangent of y divided by x.

Let arctan2(y,x) = the actual value of the arctangent, corrected for quadrant.

where x and y are the usual values in the XY rectangular coordinate plane.

Q = arctan(y/x)

if x < 0 then P = Q + pi

if x > 0 and y < 0 then P = Q + 2 pi

if x > 0 and y > 0 then P = Q

arctan2(y,x) = P

You can change it to degrees instead of radians if you want to.

The tangent of an angle is the ratio of "rise" (Y) over "run" (X). Or that of abscissa to ordinate. The arctangent really ought to begin with BOTH of those numbers (both Y and X) and return the angle. X may be either positive or negative [-1 to 1]. Y may be either positive or negative [-1 to 1].

Remember: an arctangent function does not OPERATE on an angle. It operates on a number and returns the angle. The angles which the arctangent returns might be in *any direction* (in the XY plane) starting from the point where the angle is subtended.

The single argument arctan function loses the SIGN INFORMATION of the argument, which translates into a loss of QUADRANT INFORMATION in the result. The two-argument arctangent function restores that info.

All inverse trig functions are multivalued. There really isn't much argument. This comes from the cyclic nature of the function. In fact there are some famous solutions to problems where the multivalued nature is used.

We routinely use the muliple values in notations like sin(pi/6 + 2 n pi) = 1/2 which implies: asin(1/2) = pi/6 + 2 n pi

Likewise, if arctan(y) = x is equivalent to y = tan(x) then it surely follows that is x = 10 pi giving y = 0, then a proper value for atan(0) is 10 pi.

We very often use the convention of constraining inverse trig functions to any convenient range about zero. This has nothing to do with the math or correctness of the function, just convenience. We use a similar approach to logs and roots of numbers picking only the principal values. The cube root of 1 is normally stated as 1 when -1/2 + i*sqrt(3)/2 and -1/2 - i*sqrt(3)/2 are both equally valid. They are not generally stated but must be considered in many analyses. Similarly ln(1) = 2*i*n*pi but ln(1) = 0 is usually all that is stated and is usually sufficient.

In short, there is nothing wrong with a convention that limits the range of inverse trig functions but it can be a pitfall to foget that other valid solutions exist as well.

As far as multivalued derivative to atan(x), that is not true, all solutions to atan(x) give the same derivative. They also give the same integral since the area of every 2pi cycle yields zero net area.

Edit: A note for Math_kp. The cited link, and your answer both refer top principal values. That term in itself shows that the function is multivalued. If it were not "principal" values would make no sense. The site simply assumes that the concept of principal values is understood.

As far as i can tell, multivalued function is an oximoron.

This is why the field of math prefers doing things like restricting the range of inverse trig functions. We can't take the derivative, for example, if a function has 2 values at one point.

However, there are definitely times when you want to take into account all values where the tangent value can be obtained. If some car engineer punched in inverse tan into his calculator, which spat out some random, say, 37 degree angle, when a 217 degree angle would've saved the company 5 square inches of rubber per car, life would be unhappy for this company.

Basically, anytime you do an inverse trig function, you always need to keep in mind that it IS NOT the only angle which can give you this result. often times, you need to know that htere are other solutions. you should know how to obtain the other values. the only reason we restrict the values is because of all the benefits a having a function gives us. sometimes, though, we don't care about this.

it all depends on what you want. as far as technicalities, i believe the range is only allowed to be between 90 and -90, i would simply recommend you at recognize the other angles which can get the same tangent.

It is one valued and based on link below. The principal values of inverse of fucntions are on site from wikipedia which is generally correct. If it has multiple values this is not consistant

Note: we should go by the references rather than some ones assumption when in doubt.

for example 2^2 = (-2)^2 = 4 but sqrt(4) = 2.

so we say +/- sqrt(4) and not that sqrt(4) = +/- 2

Edit:

the prerson below me mentions that arctan (x) is multivalued. I tend to disagree as

tan(y) = x can have multiple values of y for a single x but arctan(x) is the principle value and so cannot be more than one

Arctan(x) written with A capital letter means a single valued function whose range is -pi/2< Arctan x<pi/2

if you dont write it with A you must define the range

if in R

you have reason

it is difficult to explain you(i sprak arabic and frensh but not well english)

the arctang is the reverse of tang

the reverse is a function

he believe that because for many values of x tgx can have the same value!

what will he tell for the all others periodic function when we get te reverse!!

the reverse is taken in only -90 to 90 degre and not in all R

so arctg x is defioned on -90 to 90

like sqrt(x) is defined on R+

he must review his mathematics