Inverse Notation:

Caution! 

As stated at the left, the exponent of 1 denotes
"inverse". It does not mean
1/sin x,
such as x^{1} = 1/x. 

arcsin(x) =
sin^{1}(x)
arccos(x) = cos^{1}(x)
arctan(x) = tan^{1}(x) 
y = arcsin(x) = sin^{1}(x)
solves the equation x = sin(y).
Read: arcsin(x)
as "the angle whose sine is x".


When we studied inverse
functions in general (see
Inverse Functions), we learned that the inverse of a
function can be formed by reflecting the graph over the
identity line y = x. We also learned that the
inverse of a function may not necessarily be another
function.
Look at the sine function (in red)
at the right. If we reflect this function over the
identity line, we will create the inverse graph (in
blue). Unfortunately,
this newly formed inverse graph is not a function.
Notice how the green vertical
line intersects the new inverse graph in more than one
location, telling us it is not a function. (Vertical Line
Test). 

By limiting the range to see all of the
yvalues without repetition, we can define inverse
functions of the trigonometric functions.
It is possible to form inverse functions at many different locations
along the graph. The functions shown here are what are referred to
as the "principal" functions.
Inverse
sine:
f (x) = sin^{1}(x)
f (x) = arcsin(x) 
Domain: [1,1]
Range:



Inverse
cosine:
f (x) =
cos^{1}(x)
f (x)
= arccos(x) 
Domain: [1,1]
Range:



Inverse
tangent:
f (x) =
tan^{1}(x)
f (x)
= arctan(x) 
Domain:
Range:





Use the TI83+/84+
graphing calculator
to investigate trig
inverses.
Click here. 
Also see the graphs, domains
and ranges for secant, cosecant and cotangent. 

