For every trigonometry function, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. (On some calculators the arcsin button may be labelled asin, or sometimes sin-1.) So the inverse of sin is arcsin etc. When we see "arcsin A", we understand it as "the angle whose sin is A"
|sin30 = 0.5||Means: The sine of 30 degrees is 0.5|
|arcsin 0.5 = 30||Means: The angle whose sin is 0.5 is 30 degrees.|
In the above figure, click on 'reset'.
We know the side lengths but need to find the measure of angle C.
We know that so we need to know the angle whose sin is 0.5, or formally: Using a calculator to look up arcsin 0.5 we find it is 30°.
Recall that we can apply trig functions to any angle, including large and negative angles. But when we consider the inverse function we run into a problem, because there are an infinite number of angles that have the same sine. For example 45° and 360+45° would have the same sine. For more on this see Inverse trigonometric functions.
To solve this problem, the range of inverse trig functions are limited in such a way that the inverse functions are one-to-one, that is, there is only one result for each input value.
Recall that the domain of a function is the set of allowable inputs to it. The range is the set of possible outputs.
By convention, the range of arcsin is limited to -90° to +90°. So if you use a calculator to solve say arcsin 0.55, out of the infinite number of possibilities it would return 33.36°, the one in the range of the function.