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!
For a function to have an inverse, each output must have exactly one corresponding input. Thus, only onetoone functions have inverses. The tangent function doesn't have a true inverse, because the tangent function is not onetoone. So, to try and define an ‘inverse tangent function’, we do the best we can. We throw away most of the tangent curve, leaving us with a piece that has three properties:
This green piece is the restriction of the tangent curve to the interval $\,(\frac{\pi}{2},\frac{\pi}{2})\,.$ The function that the mathematical community calls ‘the inverse tangent function’ is not actually the inverse of the tangent function, because Instead, the ‘inverse tangent function’ is the inverse of this green piece of the tangent curve. 
Several Cycles of the Graph of the Tangent Function The tangent function isn't onetoone; it doesn't pass a horizontal line test. So, it doesn't have a true inverse. To define an ‘inverse tangent 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 tangent function’. 
The arctangent function (precise definition below) is the best we can do in trying to get an inverse of
the tangent function.
The arctangent function is actually the inverse of the green piece shown above!
Here's a ‘function box’ view of what's going on:
The tangent function takes a real number
(excluding $\,\frac{\pi}{2} + k\pi\,$ for integers $\,k\,$)
as an input.
It gives a real number output. For example (as below), the output $\,0.5\,$ might come from the tangent function. 
When we try to use the tangent 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 tangent 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 $\,(\frac{\pi}2,\frac{\pi}2)\,.$ 
It's a bit of a misnomer, but the arctangent function (precise definition below) is often referred to
as the ‘inverse tangent function’.
A better name would be something like ‘the inverse of an appropriatelyrestricted tangent function’.
(It's no surprise, however, that people don't say something that long and cumbersome.)
So, what exactly is $\,\arctan 0.5\,$?
More generally, let $\,x\,$ be any real number.
Then:
In my own mind (author Dr. Carol Burns speaking here), the words I say are:
The precise definition of the arctangent function follows.
It can look a bit intimidating—the notes following the definition should help.
Here's the piece of the tangent curve that is used to define the arctangent function: domain: $\,(\frac{\pi}{2},\frac{\pi}{2})\,$ range: $\,(\infty,\infty)\,$ 
Here's the same curve, together with its reflection about the line $\,\color{red}{y = x}\,$ 
The graph of the arctangent function domain: $\,(\infty,\infty)\,$ range: $\,(\frac{\pi}{2},\frac{\pi}{2})\,$ 
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 arctangent function, then apply the tangent function,
and end up right where you started.
Here are the details:
For all $\,x\in \Bbb R\,,$
$$
\cssId{sb65}{\tan(\arctan x) = x}
$$


Here's the direction where they don't necessarily ‘undo’ each other nicely:
start with a number, first apply the tangent function, then apply the arctangent function.
If the number you started with is outside the interval $\,(\frac{\pi}2,\frac{\pi}2)\,,$
then you don't end up where you started! Here are the details: For all $\,x\in (\frac{\pi}2,\frac{\pi}2)\,,$ $$ \cssId{sb75}{\arctan(\tan x) = x} $$ For all $\,x\,$ in the domain of the tangent function but not in $\,(\frac{\pi}2,\frac{\pi}2)\,,$ $$ \cssId{sb78}{\arctan(\tan x) \ne x} $$ 



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