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.

Figure %: The domains of 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.

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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.