Graphs of the Inverse Trig Functions
Topic Index | Algebra2/Trig Index | Regents Exam Prep Center


 

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 y-values 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 TI-83+/84+
graphing calculator
to investigate trig
inverses.
Click here.

Also see the graphs, domains and ranges for secant, cosecant and cotangent.