You DO things.
In math (as in life), only sometimes can you ‘undo’ what you did.
Sometimes you can't fully or reliably ‘undo’ something, but you do the best you can.
In math, functions are a common tool to do things.
You feed an input into a function.
The function does something to the input to produce a unique output.
For example, feed $\,x\,$ into the sine function (‘$\,\sin\,$’, pronounced ‘sine’—long i) and the unique output ‘$\,\sin x\,$’ (sine of $\,x\,$) results, as shown at right. Recall that $\,\sin x\,$ (no parentheses) is a shorthand for $\,\sin(x)\,$ (with parentheses). For multiletter function names, you can drop the parentheses that hold the input, as long as there is no confusion about order of operations. 
the sine function: input $\,x\,$ gives output $\,\sin x\,$ 
Going from inputs to outputs is the most common way to use functions. However, sometimes you need to (try and) use functions ‘backwards’. You want to start with an output and try to get back to an input. Functions do something; you want to try and undo what the function did. When you try to go from outputs to inputs, you're in the realm of inverse functions. The questions to be explored in this section are:

trying to use the sine function ‘backwards’ 
To illustrate the idea, suppose there's an unknown number $\,x\,$. All we know about $\,x\,$ is that it's a real number, and when it gets dropped into the sine function, the number $\,0.5\,$ is the corresponding output. Thus, we know that $\,\sin x = 0.5\,$. Does this output information uniquely identify the input? That is, does knowing ‘$\,\sin x = 0.5\,$’ uniquely identify $\,x\,$? 

The graph of the sine function (at right) shows that the answer is a resounding NO! All the values of $\,x\,$ shown (and infinitely many more, by continuing the pattern to the right and left) satisfy $\,\sin x = 0.5\,$. Any of these values of $\,x\,$ could have been dropped into the sine function to give the output $\,0.5\,$. Without additional information, there's no way to know precisely which value of $\,x\,$ produced the output $\,0.5\,$. 

For the sine function, an output does not uniquely identify an input.
For the sine function, any output (between $\,1\,$ and $\,1\,$) has infinitely many corresponding inputs.
The graph of the sine function does not pass a horizontal line test.
The sine function is not onetoone.
The same problem holds with all the trigonometric functions: sine, cosine, tangent, cotangent,
secant, and cosecant.
None of them are onetoone.
None of them have true inverses.
This is so important, that it is worth repeating:
Feeling a bit rusty on onetoone functions and inverse functions?
If so, you may want to review the prior sections listed below before continuing:
For a function to have an inverse, it must be onetoone.
Trigonometric functions aren't onetoone, so they don't have true inverses.
However, we still want to be able to go from any output to a corresponding input;
it's just that we'll have to choose one particular input from the infinitelymany that are available.
To accomplish this, we'll throw away most of the graph,
leaving just a piece that provides us with exactly one input for each of the possible outputs.
More precisely, we need to restrict the domain of the original trigonometric function.
We want a ‘restricted version’ that has these three properties:
To better understand these three conditions,
let's investigate the potential ‘restricted versions of the sine function’ suggested below.
For each, answer these questions:










The interval $\,[\frac{\pi}{2},\frac{\pi}{2}]\,$ is the only interval that satisfies all three conditions! Thus, restriction to the interval $\,[\frac{\pi}{2},\frac{\pi}{2}]\,$ is used to define the inverse sine function. Given any output (say, $\,0.5\,$) from the sine function, there are infinitely many inputs $\,x\,$ for which $\,\sin x = 0.5\,$. However, there is only one value of $\,x\,$ between $\,\frac{\pi}{2}\,$ and $\,\frac{\pi}{2}\,$ that works! 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
