Question 807759
ANSWER:
The function {{{arc}}}{{{cos(x)}}} is defined as the angle {{{theta}}} that satisfies {{{cos(theta)=x}}} and {{{0<=theta<=pi}}}
(and we measure angles in radians, not degrees).
If the question is finding the angle {{{theta}}} such that {{{cos(theta)=1}}}, 
then there are infinite answers, 
so you have to give them as {{{theta =2k*pi}}} in radians,
(or {{{theta =k*360^o}}} if you measure angles in degrees),
where {{{k}}} is any integer.
However, to define {{{arc}}}{{{cos(x)}}} as a function,
we choose just the most trivial of those answers, {{{theta=0}}} ,
so {{{arc}}}{{{cos(1)=0}}} (just one answer).
 
EXPLANATION AND MORE:
The drawing below shows the unit circle and an angle, {{{theta=AOB}}} , in standard position.
{{{drawing(300,300,-1.2,1.3,-1.2,1.3,
grid(0),
blue(circle(0.866,0.5,0.03)),locate(0.9,0.55,B),
blue(circle(0,0,0.03)),locate(0.02,0,O),
blue(circle(1,0,0.03)),locate(1.02,0.15,A),
red(circle(0,0,1)),
red(arrow(0,0,1.126,0.65)),
arc(0,0,0.6,0.6,-30,0),locate(0.3,0.15,theta),
green(circle(-0.866,0.5,0.03)),locate(-0.95,0.55,D),
green(circle(0.866,-0.5,0.03)),locate(0.9,-0.45,C)
)}}} {{{theta =30^o}}} or {{{theta=pi/6}}} depending on the units you use for angles.
Angles are measured as turns counterclockwise from ray OA to ray OB.
It is a useful way to define angles for technical issues,
such as stating the angle a control knob has to be turned to get to the desired position.
You start with OB coinciding with OA and {{{theta=0}}}.
As you turn OB around O, you eventually go one full turn and get back to OA,
and that turn is considered to be {{{360^o}}} or {{{2pi}}}.
After another turn, the angle would be {{{720^o}}} or {{{4pi}}}
I say that the angle {{{theta}}} in my drawing is {{{30^o}}} ,
because I say that I just turned from OA to OB the shortest possible way.
If I said that I turned a little over 2 turns clockwise on my way from OA to OB, then it would be
{{{theta=720^o+30^o=750^o}}}
Cosine is defined as the x-coordinate of point B, while sine is the y-coordinate.
In my drawing, those coordinates are
{{{x[B]=cos(theta)=sqrt(3)/2=about}}}{{{0.866}}} and
{{{y[B]=sin(theta)=1/2=0.5}}} .
There is another point in the circle (point C) with the same x-coordinate,
as there is another point (point D) with the same y-coordinate.
 
It is obvious that the only point in the circle with {{{x=1}}} is point A,
so for {{{cos(theta)=1}}} we get infinite answers, but just one answer per turn,
and the answers are all {{{2pi}}} or {{{360^o}}} apart.
The same thing happens for {{{x=-1}}} and {{{arc}}}{{{cos(-1)}}} at the other extreme of the range of {{{x}}} values for the circle.
For values in between those extremes, such as
{{{sqrt(3)/2=about}}}{{{0.866}}} ,
there are twice as many answers,
because there are twice as many points with that x-coordinate.
{{{x[B]=sqrt(3)/2=x[C]}}} so {{{cos(AOB)=sqrt(3)/2=cos(AOC)}}}
and we can say {{{AOC=-30^o}}} or {{{AOC=-pi/6}}} if we choose the simplest way to define those angles.
For the infinite answers to {{{cos(theta)=sqrt(3)/2}}} ,
we have to express all the coterminal angles (that differ from AOB and AOC by a whole number of turns) as
{{{theta=2k*pi +- pi/6}}} (or {{{theta=k*360^o +- 30^o}}} if we use degrees) for any integer {{{k}}} .