The inverse trigonometric relations are not functions because for any given input there exists more than one output. That is, for a given number there exists more than one angle whose sine, cosine, etc., is that number. The ranges of the inverse relations, however, can be restricted such that there is a one-to-one correspondence between the inputs and outputs of the inverse relations. With these restricted ranges, the inverse trigonometric relations become the inverse trigonometric functions.
The symbols for the inverse functions differ from the symbols for the inverse relations: the names of the functions are capitalized. The inverse functions appear as follows: Arcsine, Arccosine, Arctangent, Arccosecant, Arcsecant, and Arccotangent. They can also be represented like this: y = sin-1(x), y = cos-1(x), etc. The chart below shows the restricted ranges that transform the inverse relations into the inverse functions.
The inverse trigonometric functions do the same thing as the inverse trigonometric relations, but when an inverse functions is used, because of its restricted range, it only gives one output per input--whichever angle lies within its range. This creates a one-to-one correspondence and makes the inverse functions more usable and useful.
Knowledge of Trigonometric and Inverse Trigonometric Functions Brings Great Power (and great responsibility)
With knowledge of the trigonometric functions, we can calculate the value of a function at a given angle. With the inverse trigonometric functions, we can now calculate angles given certain function values. Solving both ways will be especially helpful as we attempt to solve triangles in the upcoming sections.