Precalculus
Houghton Mifflin, Fifth Edition
Chapter 4
Section 7
Inverse Trig Functions
Arcsine
Arccosine and Arctangent
Arcsine Function
The Arcsine
function is defined
as the inverse of
the sine function.
It is shown to the
right.
Inverse Sine Graph
You know that a function’s
inverse is its reflection across
the line y = x.
Each point (x, y) on the graph
of a function is matched to a
point (y, x) on the graph of the
inverse.
The point (π, 0) is on the sine
graph, and the point (0, π) is
on the graph of its inverse.
By this definition, the inverse
sine graph would look like the
blue graph to the right.
(π, 0)
(0, π)
Arcsine Function
You will note, however, that the inverse
sine graph is NOT a function. It fails
the vertical line test miserably.
We would like to be able to refer to a
function that is the inverse of the sine
function, though, so we define the
arcsine function to be only a portion of
the inverse sine graph.
You will note that although vertical
lines will pass through the inverse sine
graph multiple times, each vertical line
only crosses the bold section of the
graph exactly once.
This is the segment where the y-values
lie between –π/2 to π/2.
Also note that if this segment is made
any longer, there will be a vertical line
that crosses the segment more than
once. This segment is as long as it
can possibly be without failing the
vertical line test.
Arcsine Domain and Range
Does this segment cover all of the
domain of the sine function?
We noted that this segment has a
domain [-1, 1] and a range [–π/2,
π/2]. Thus, the inverse of this
function, the sine function, must
have a range [-1, 1] and a domain
[–π/2, π/2] since each point (x, y)
on one graph must be matched to
a point (y, x) on its inverse. This
means that the x- and y-values
are switched between functions
that are each other’s inverses.
Therefore only the section of the
sine function from [–π/2, π/2] has
an inverse.
Arccosine and Arctangent
Similarly, we must restrict the arccosine and
arctangent graphs to obtain functions.
The portions shown in bold are the functions f(x) =
arccos(x) and g(x) = arctan(x).