Recall that if the graph of an equation passes the Vertical Line Test (ie that a vertical line anywhere on the graph touches the graph in at most one point), then it is a special type of graph which we call a function.

Now, not all functions are created equal. Some functions are even more special because they have inverese ("undoing") functions. To tell if a function has an inverese (ie that it is a one-to-one function) we use the Horizontal Line Test. The Horizontal Line Test is just like the Vertical Line Test with the obvious exception: if you can place a HORIZONTAL line anywhere on the graph of the function and it touches the graph in at most one place, then the function is one-to-one = has an inverse function. If any horizontal line touches the graph in more than one point, then the original function is not one-to-one and will not have an inverse (unless we do something special we'll talk about in a minute).

You may recall from trigonometry that the sine function repeats itself forever

so it can't pass the Horizontal Line Test. We see this in the next figure where there are two horizontal lines (one orange, one red) that each touch the graph in more than one point.

Since y = sin(x) is not one-to-one we restrict the domain of the sine function so that it will pass the Horizontal Line Test (and have an inverse). A natural choice is from -π/2 to π/2, including the endpoints. (See the graph below).

At this point you may wish to view the video clip The definition of the inverse sine function (12:40) before going on.

When dealing with the sine function (or any other trig function for that matter) we put an angle into the function and the answer is a number.
For example: sin(30°) = 0.5. An angle goes in and a number comes out. Therefore, the inverse sine function, denoted arcsin or sin^-1, does just the opposite: we put in a number and it gives an answer that is an angle.
For example: arcsin(0.5) = 30° or arcsin(0.5) = π/6. We could also use the inverse notation: sin^-1(0.5) = 30° or sin^-1(0.5) = π/6. The two notations, arcsin and sin^-1 , are interchangeable and the answer may be given in either degrees or radians.

We could continue finding values of arcsin(x) and we would get the following table:

Another way we could come up with the graph of the arcsine function is to realize that it is the inverse of the sine function. You will recall that the graph of the inverse function is the mirror image over the line y = x. Either using the table or the inverse relationships, we see that the graph of y = sin^-1(x) = arcsin(x) looks like the purple curve in the following figure.

Now, before we go any further, I need to caution you about a mistake too many students make regarding notation. While it is true that sin²(x) = (sin(x))² the -1 superscript on a function DOES NOT mean the reciprocal of the function. In other words sin^-1(x) ≠ 1/sin(x). Recall that the reciprocal of the sine function is the cosecant function: 1/sin(x) = csc(x).

Now that we understand more about the inverse sine function, here's the best way that I know how to explain it in a way to remember it. If y = sin(x) then we stick the angle, whose value is x, into the sine function and that gives us a number for the value of y.
For example, if y = sin(π/4) then x is the angle (x =π/4 radians) that goes into the function (in this case the function is sine) and y is the number from -1 to 1 that we get out (namely sqrt(2)/2). Again we put an angle in and got a number out.

Now if we have y = arcsin(x), which is the same as saying y = sin^-1(x), things are different. You need to recognize that this time x is a number and y is the angle because we are dealing with the arcsine function (= inverse sine function) and it undoes the sine function.

Let's take the example y = sin^-1(sqrt(2)/2). Here, x is a number from -1 to 1 (namely, x = sqrt(2)/2) and y is the angle (that we are trying to find).
Next we think, "The sine of what angle, y, will give me sqrt(2)/2?" or in mathematics, sin(y) = sqrt(2)/2.
Here's where your memorizing the important angles becomes crucial. If you've been diligent, you will remember that sin(45°) = sin(π/4) = sqrt(2)/2.
Comparing these two equations, sin(y) = sqrt(2)/2 and sin(π/4) = sqrt(2)/2, we easily see that y = π/4 radians = 45°.

Let's conclude this part of the lesson using a table that will help emphasize the inverse relationship of the sin and arcsin functions.