Before studying this section, you are encouraged to read Trying to ‘Undo’ Trigonometric Functions.
For a function to have an inverse, each output must have exactly one corresponding input.
Thus, only onetoone functions have inverses.
The sine function doesn't have a true inverse, because the sine function is not onetoone.
So, to try and define an ‘inverse sine function’, we do the best we can, as discussed below.
To try and define an ‘inverse sine function’, proceed as follows: Throw away most of the sine curve, leaving us with a piece that has three properties:
This green piece is the restriction of the sine curve to the interval $\,[\frac{\pi}{2},\frac{\pi}{2}]\,$. The function that the mathematical community calls ‘the inverse sine function’ is not actually the inverse of the sine function, because Instead, the ‘inverse sine function’ is the inverse of this green piece of the sine curve. 
Several Cycles of the Graph of the Sine Function The sine function isn't onetoone; it doesn't pass a horizontal line test. So, it doesn't have a true inverse. To define an ‘inverse sine 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 sine function’. 
The arcsine function (precise definition below) is the best we can do in trying to get an inverse of
the sine function.
The arcsine function is actually the inverse of the green piece shown above!
Here's a ‘function box’ view of what's going on:
The sine 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 sine function. 
When we try to use the sine 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 sine 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 arcsine function (precise definition below) is often referred to
as the ‘inverse sine function’.
A better name would be something like ‘the inverse of an appropriatelyrestricted sine function’.
(It's no surprise, however, that people don't say something that long and cumbersome.)
So, what exactly is $\,\arcsin 0.5\,$?
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 arcsine function follows.
It can look a bit intimidating—the notes following the definition should help.
Here's the piece of the sine curve that is used to define the arcsine function: domain: $\,[\frac{\pi}{2},\frac{\pi}{2}]\,$ range: $\,[1,1]\,$ 
Here's the same curve, together with its reflection about the line $\,\color{red}{y = x}\,$ 
The graph of the arcsine function domain: $\,[1,1]\,$ 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 arcsine function, then apply the sine function,
and end up right where you started.
Here are the details: For all $\,x\in [1,1]\,$, $$ \cssId{sb67}{\sin(\arcsin x) = x} $$


Here's the direction where they don't necessarily ‘undo’ each other nicely:
start with a number, first apply the sine function, then apply the arcsine 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 $\,x\in [\frac{\pi}2,\frac{\pi}2]\,$, $$ \cssId{sb77}{\arcsin(\sin x) = x} $$ For all $\,x\not\in [\frac{\pi}2,\frac{\pi}2]\,$, $$ \cssId{sb80}{\arcsin(\sin x) \ne x} $$ 



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