inverse trigonometric functions
The inverse trigonometric functions are ,
and
. These functions perform the reverse operations to the original trigonometric functions
,
and
respectively. Recall that a function is invertible if it is one-to-one. Click here to revise inverse functions. Hence, before we can sketch the graphs of the inverse trigonometric functions, we must choose a domain for them for which they are one-to-one. Note that the original trigonometric functions work on angles and so each of the inverse trigonometric functions will return an angle. We use radians for all angles in the following – see more on radians. Also note that we use
instead of
, for example, as this can be confused with
, the reciprocal trigonometric function.
Inverse Trigonometric Function: arcsin(x)
Since is periodic, there are infinitely many regions for which it is one-to-one. We choose the default domain to be
. The range of
is
. It follows that the domain of
is
and the range is
. The graphs of
and
are reflections of one another in the line
.
Inverse Trigonometric Function: arccos(x)
Since is periodic, there are infinitely many regions for which it is one-to-one. We choose the default domain to be
. The range of
is
. It follows that the domain of
is
and the range is
. The graphs of
and
are reflections of one another in the line
.
Inverse Trigonometric Function: arctan(x)
Since is periodic, there are infinitely many regions for which it is one-to-one. We choose the default domain to be
. The range of
is all of
. It follows that the domain of
is
and the range is
. The graphs of
and
are reflections of one another in the line
.