From my understanding, the atan2() function exists in programming languages because atan() itself cannot always determine the correct theta since the output is restricted to -pi/2 to pi/2.
If this is the case, then the same problem applies to both asin() and acos(), both of whom also have restricted ranges, so then why are there no asin2() and acos2() functions?
First off, note that the syntaxes of the two arctan functions are atan(y/x) and atan2(y, x). This distinction is important, because by not performing the division you provide additional information, most importantly the individual signs of x and y. If you know the individual x and y coordinates, the particular solution to the atan function can be found (i.e. the solution which takes into account the quadrant that (x,y) is in).
If you go from tan(θ) = y/x to sin(θ) = y/sqrt(x²+y²), then the inverse operation asin takes y and sqrt(x²+y²) and combines that to obtain some information about the angle. Here it doesn't matter whether we perform the division ourself or let some hypothetical asin2 function handle it. The denominator is always positive, so the divided argument contains just as much information as separate numerator and denominator contain. (At least in an IEEE environment where division by zero leads to a correctly-signed infinity.)
If you know the y coordinate and the hypothenuse sqrt(x²+y²) then you know the sine of the angle, but you cannot know the angle itself, since you cannot distinguish between negative and positive x values. Likewise, if you know the x coordinate and the hypothenuse, you know the cosine of the angle but you cannot know the sign of the y value.
So asin2 and acos2 are not mathematically feasible, at least not in an obvious way. If you had some kind of sign encoded into the hypothenuse, things might be different, but I can think of no situation where such a sign would arise naturally.
Because asin(y,x) acos(y,x) would each take the same parameters as atan(y,x) and each give the same answer. Each would be equally valid, but we only need one such function.
The unclarity arises from the name (of atan2). Its a function that given x and y, computes the angle (made by a line from the origin to this point) with the (positive) x-axis. A name like angle_from(x,y) would arguably have been more appropriate.
I will explain in SIMPLE TERMS this way.
Refer to this image for the following explanation:
Task: Choose a function that will track the correct angle across a range -180 < θ < 180
Trial 1:
sin() is positive in the first and second quadrants, sin(30) = sin(150) = 0.5. It won't be easy to track quadrant change with sin().
Therefore, asin2() is not feasible.
Trial 2:
cos() is positive in the first and fourth quadrants, cos(60) = sin(300) = 0.5. Also, it won't be easy to track quadrant change with cos().
Therefore, acos2() is again not feasible.
Trial 3:
tan() is positive in the first and third quadrants, and in an interesting order.
It is positive in the 1st quadrant, negative in the 2nd, positive in the 3rd, negative in the 4th, and positive in the wrapped-around-1st quadrant.
such that tan(45) = 1 , tan(135) = -1, tan(225) = 1, tan(315) = -1, and tan(360+45) = 1. Hurray! we can track quadrant change.
Notice that the unambiguous range is -180 < θ < 180. Also, note in my 45-degree-increment example above, if the sequence is 1,-1,.. the angle goes counter-clockwise, and if the sequence is -1,1,.. it goes clockwise. This idea should resolve directionality.
Therefore, atan2() BECOMES OUR CHOICE.
There are times when a function like "acos2" is needed, for example when performing rotations of vectors in 3D space. Under those circumstances, I hard-code my own acos2 function which simply performs the following checks:
x_perp=sqrt(x*x+y*y)
r=sqrt(x*x+y*y+z*z)
if(x_perp.gt.0.0d0) then
phi=acos(x/x_perp)
else
phi=0.0d0
endif
if(y.lt.0.0d0) phi=2.0d0*pi-phi
theta=acos(z/r)
where theta and phi are the usual spherical coordinates and x,y,z the Cartesian coordinates. The problem arises when y is negative, there needs to be a phase shift in phi. There is no such problem for theta.
