how to define arc cosine(cosine x)

Question

for -360 deg<x< 360 deg??
a proper conceptual approach to the solution will be appretiated..

The function Cos(x) is equal to 1 at x = 0 and it decreases as a function of x until x = pi (=180 degrees). There the cosine is minus one.

So, all the values the cosine function can reach are also reached in the interval from x = 0 to x = 180 degrees. All the values are reached exactly once in this interval, so for each possible value the cosine can reach there is a unique x in the interval from zero to 180 degrees such that cos(x) equals that value.

The arcos function is defined as follows:

If x is between zero and pi (= 180 degrees) then:

arccos(cos(x)) = x

If x is between minus 180 degrees and zero then, because cos(x) = cos(-x) and -x is in the range between zero and 180 degrees you have:

arccos(cos(x)) = arccos(cos(-x)) = -x

If x is between 180 degrees and 360 degrees, then because
cos(x) = cos(x - 360) and x-360 is between minus 180 and zero degrees we can use the above formula:

arccos(cos(x)) = arccos(cos(x-360)) = 360-x

If x is between minus 360 degrees and minus 180 degrees you can use:

arccos(cos(x)) = arccos(cos(-x))

-x is in the range between 180 and 360 degrees and we looked at this case above. We thus have:

arccos(cos(x)) = 360 + x

thanks count iblis
this was just the answer i was looking for.
u r a genius
so simple.. y wasn't i thinking of that