Before studying this section, you are encouraged to read
Trying to ‘Undo’ Trigonometric Functions.
This section is a copy of Inverse Trigonometric Function: Arcsine, with appropriate changes.
If you've mastered the arcsine section, then this one should be quick and easy!
The arccosine function was informally introduced in Using the Law of Cosines in the SSS case (& Introduction to the Arccosine Function).
It is made precise here.
For a function to have an inverse, each output must have exactly one corresponding input. Thus, only onetoone functions have inverses. The cosine function doesn't have a true inverse, because the cosine function is not onetoone. So, to try and define an ‘inverse cosine function’, we do the best we can. We throw away most of the cosine curve, leaving us with a piece that has three properties:
Positive numbers are easier to work with than negative numbers. This green piece is the restriction of the cosine curve to the interval $\,[0,\pi]\,$. The function that the mathematical community calls ‘the inverse cosine function’ is not actually the inverse of the cosine function, because Instead, the ‘inverse cosine function’ is the inverse of this green piece of the cosine curve. 
Several Cycles of the Graph of the Cosine Function The cosine function isn't onetoone; it doesn't pass a horizontal line test. So, it doesn't have a true inverse. To define an ‘inverse cosine function’, we do the best we can. Throw away most of the curve— leave only the green part. This green part is onetoone. This green part does have an inverse. The inverse of this green part is what the mathematical community calls ‘the inverse cosine function’. 
The arccosine function (precise definition below) is the best we can do in trying to get an inverse of
the cosine function.
The arccosine function is actually the inverse of the green piece shown above!
Here's a ‘function box’ view of what's going on:
The cosine function takes a real number as an input. It gives an output in the interval $\,[1,1]\,$. For example (as below), the output $\,0.5\,$ might come from the cosine function. 
When we try to use the cosine function box ‘backwards’,
we run into trouble.
The output $\,0.5\,$ could have come from any of the inputs shown. 
However,
when we use the green piece of the cosine curve, the problem is solved! Now, there's only one input that works. (It's the value of the green $\,\color{green}{x}\,$.) Observe that $\,\color{green}{x}\,$ is in the interval $\,[0,\pi]\,$. 
It's a bit of a misnomer, but the arccosine function (precise definition below) is often referred to
as the ‘inverse cosine function’.
A better name would be something like ‘the inverse of an appropriatelyrestricted cosine function’.
(It's no surprise, however, that people don't say something that long and cumbersome.)
So, what exactly is $\,\arccos 0.5\,$?
More generally, let $\,x\,$ be any number in the interval $\,[1,1]\,$.
Then:
In my own mind (author Dr. Carol Burns speaking here), the words I say are:
The precise definition of the arccosine function follows.
It can look a bit intimidating—the notes following the definition should help.
Here's the piece of the cosine curve that is used to define the arccosine function: domain: $\,[0,\pi]\,$ range: $\,[1,1]\,$ 
Here's the same curve, together with its reflection about the line $\,\color{red}{y = x}\,$ 
The graph of the arccosine function domain: $\,[1,1]\,$ range: $\,[0,\pi]\,$ 
Notice that the domain and range of a function and its inverse are switched! The domain of one is the range of the other. The range of one is the domain of the other. 
Here's the direction where they do ‘undo’ each other nicely:
start with a number, first apply the arccosine function, then apply the cosine function,
and end up right where you started.
Here are the details: For all $\,x\in [1,1]\,$, $$ \cssId{sb66}{\cos(\arccos x) = x} $$


Here's the direction where they don't necessarily ‘undo’ each other nicely:
start with a number, first apply the cosine function, then apply the arccosine function.
If the number you started with is outside the interval $\,[0,\pi]\,$,
then you don't end up where you started! Here are the details: For all $\,x\in [0,\pi]\,$, $$ \cssId{sb76}{\arccos(\cos x) = x} $$ For all $\,x\not\in [0,\pi]\,$, $$ \cssId{sb79}{\arccos(\cos x) \ne x} $$ 



On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
