the cosine of pi is -1. the inverse cosine function (arccosine) of -1 will obviously give us pi.
The cosine of (3*pi) is -1 as well 😉.
Explanation:
A relationship is something that maps a set of values (a,b,c,d) onto their images (R(a),R(b),R(c),R(d)). The inverse relationship is the one that maps R(x) to x for all image-points.
If every value in the domain (the input, this case a,b,c,d) has only one image, then we call the relationship a function.
However, the inverse relationship of a function is not always (in fact most of the time) a function as well. For example the relationship f(x)=x^2 for all real points is a function. The inverse relationship (y where y^2=value) has to map for example '4' to '2' and '-2', so it is not a function.
On the other hand, you can split the inverse relationship up into two inverse functions sqrt(x) and -sqrt(x) and pick one of them to use (we normally use the positive one).
Similarly, look at the cosine function. For every image-value between -1 and 1, there are an infinite number of points in the domain that map to them, each offset by 2*pi. For example:
...
cos(-2*pi+pi) = -1
cos(pi) = -1
cos(2*pi+pi) = -1
cos(4*pi+pi) = -1
...
So the inverse relationship is definitely not a function, and you can't split it up into functions, because there are an infinite amount of them.
So what they do is take one of those functions (for acos it is the interval [0,pi]) and use that to define the inverse function 😉