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Mathematical Expression Editor
Motivating Questions
For the restricted sine, tangent, and secant functions, how do we define
the corresponding arcsine, arctangent, and arcsecant functions?
What are the key properties of arcsine, arctangent, and arcsecant?
Introduction
In the last section we defined arccosine, the inverse for cosine restricted to a single
period. In this section we will explore the definition of similar inverse functions on
restricted domains of sine, tangent, and secant.
As we recalled last time,
A function has an inverse function if and only if there exists a function
that undoes the work of : that is, there is some function , the inverse of ,
for which for each in the domain of , and for each in the range of .
A function has an inverse function if and only if the graph of passes the
Horizontal Line Test.
When has an inverse, we know that “” and “” are two different
perspectives on the same statement.
As with the cosine function, the trigonometric functions , , and are periodic, so they
fail the horizontal line test. Hence, considering these functions on their full domains,
neither has an inverse function. At the same time, it is reasonable to think about
changing perspective and viewing angles as outputs in certain restricted settings, as
we did with cosine.
The Arcsine Function
We can develop an inverse function for a restricted version of the sine function in a
similar way. As with the cosine function, we need to choose an interval on which the
sine function is always increasing or always decreasing in order to have the function
pass the horizontal line test. The standard choice is the domain on which is
increasing and attains all of the values in the range of the sine function. Thus, we
consider so that has domain and range and use this restricted function to
define the corresponding arcsine function.
Let be defined on the domain , and
observe has the range . For any real number that satisfies , the arcsine of ,
denoted
is the angle satisfying such that . Note that we use interchangeably with with
.
The goal of this activity is to understand key properties of the arcsine function in a
way similar to our discussion of the arccosine function in the previous section. We
will use our deductive reasoning skills a la Sherlock Holmes to build off our discussion
from the last section.
(a)
Using the definition of arcsine given above, what are the domain and range of
the arcsine function?
The domain of arcsine is .
The range of arcsine is .
(b)
Determine the following values exactly:
(c)
On the axes provided below, sketch a careful plot of the restricted sine function
on the interval along with its corresponding inverse, the arcsine function. Label
at least three points on each curve so that each point on the sine graph
corresponds to a point on the arcsine graph. In addition, sketch the line to
demonstrate how the graphs are reflections of one another across this
line.
(d)
True or false: ? truefalse Write a complete sentence to explain your reasoning.
Properties of the arcsine function.
The restricted sine function, , is defined on the domain with range . This
function has an inverse function that we call the arcsine function, denoted
.
The domain of is with range .
The arcsine function is always increasing on its domain.
Below we have a plot of the restricted sine function (in light blue) and its
corresponding inverse, the arcsine function (in dark blue).
Exploring Arcsine
Let’s solve the following equations analytically, then we can consider the graph of
arcsine.
(a)
We start by finding . Remember that for in , is the value in such that
.
Hence, is , and we now see that
Now, if you’re thinking, “Hey, we didn’t need that extra step!” Then you would
be correct. But why didn’t we need that final step?
Let’s recall how we defined arcsine. Since sine is a periodic function, it fails the
horizontal line test. However, if we restrict sine to a portion of its domain on
which it is only increasing, , then we may define a function on this domain
such that for in . Arcsine then is defined as the inverse of this function .
Therefore, is the inverse of arcsine. Thus, in practice, sine is the inverse of
arcsine.
A word of caution: As was the case with arccosine and cosine, arcsine is
only the inverse of restricted sine. We will illustrate this with the next
example.
(b)
It may be tempting to take a look at this expression and conclude that the
solution is since arcsine is the inverse of sine.
Hold those horses!
Remember, we had to restrict the domain of sine in order to define an inverse
function, which we called arcsine. Arcsine is the inverse of the restricted sine
function, whose domain is . is larger than , so it is not within the domain of
this restricted sine function.
Thus, we begin by simplifying .
Now, let’s consider , recalling again the range of arcsine. We are looking for the
value of in such that .
Hence, is , and we now see that
(c)
First, we observe that is in the range of arcsine, so there should be a solution.
We will now use the fact that sine is the inverse of arcsine to reduce this to a
linear equation.
Thus, we have
which is equivalent to .
The Arctangent Function
Next, we develop an inverse function for a restricted version of the tangent
function. We choose the domain on which is increasing and attains all of the
values in the range of the tangent function.
Let be defined on the domain ,
and observe has the range . For any real number , the arctangent of ,
denoted
is the angle satisfying such that . Note that we use interchangeably with with
.
Let us once again channel our inner Sherlock Holmes to understand key properties of
the arctangent function.
(a)
Using the definition given above, what are the domain and range of the
arctangent function?
The domain of arctangent is .
The range of arctangent is .
(b)
Determine the following values exactly:
.
(c)
The restricted tangent function on the interval is plotted below. On the same
axes, sketch its corresponding inverse function (arctangent). Label at least three
points on each curve so that each point on the tangent graph corresponds
to a point on the arctangent graph. In addition, sketch the line to
demonstrate how the graphs are reflections of one another across this
line.
(d)
Complete the following sentence: “as increases without bound, ” increases
without bounddecreases without boundincreases toward decreases toward
Properties of the arctangent function.
The restricted tangent function, , is defined on the domain with range .
This function has an inverse function that we call the arctangent function,
denoted .
The domain of is with range .
The arctangent function is always increasing on its domain.
Below we have a plot of the restricted tangent function (in light blue) and
its corresponding inverse, the arctangent function (in dark blue).
Exploring Arctangent
Let’s solve the following equations analytically, then we can consider the graph of
arctangent.
(a)
We start by finding . Remember that for in , is the value in such that
.
Hence, is , and we now see that
Now, I know you’re thinking, “Hey, why do you keep making us do an extra
step?” It’s because it is imperative that you consider the range of the arc
trig functions. These are considerations you will also need to make when we
start combining different trig functions with the inverses of others (say sine of
arctangent of a value).
Let’s recall how we defined arctangent. Since tangent is a periodic function, it
fails the horizontal line test. However, if we restrict tangent to a single period
(note tangent only increasing), , then we may define a function on this domain
such that for in . Arctangent then is defined as the inverse of this function .
Therefore, is the inverse of arctangent. Thus, in practice, tangent is the inverse
of arctangent.
A word of caution: As was the case with the previous two trig functions and
their respective inverses, arctangent is only the inverse of restricted tangent. We
will illustrate this with the next example.
(b)
It may be tempting to take a look at this expression and conclude that the
solution is since arctangent is the inverse of tangent.
But, wait!
Remember, we had to restrict the domain of tangent in order to define an
inverse function, which we called arctangent. Arctangent is the inverse of the
restricted tangent function, whose domain is . is larger than , so it is not within
the domain of this restricted tangent function.
Thus, we begin by simplifying .
Now, let’s consider , recalling again the range of arctangent. We are looking for
the value of in such that .
Hence, is , and we now see that
(c)
We will treat this like a quadratic equation to begin, as we did in the section
Some Applications of Trig Functions.
Let , then we have a standard quadratic equation: . Factoring, we see that this
is equivalent to
This has two solutions: and . In other words, we now simply solve (a) and (b)
. is not in the range of arctangent, so (b) does not have a solution. Hence, this
cannot be a solution to our equation, and we must look at (a). is in the range
of arctangent, so the solution to (a) will be a solution to our original
equation.
Since tangent is the inverse to arctangent, the equation (a) is equivalent
to
which is further equivalent to
The Arcsecant Function
We will also consider the inverse function for a restricted version of the secant
function. As with the cosine and sine functions, we need to choose an interval on
which the secant function is always increasing or always decreasing in order to have
the function pass the horizontal line test. In the case of secant, this means choosing
two distinct intervals. A word of caution in working with the restricted secant
function and its associated inverse, there is not a “standard” choice for the
domain of restricted secant. However, we will establish a convention in this
course.
We restrict the domain of the function to , where secant is increasing on each
interval and attains all the values within the range of the secant function. By
reflecting across the line and switching the and coordinates we are able to define
the function as follows.
Let be defined on the domain , and observe that has the range . For any real
number , the arcsecant of y, denoted
is the angle satisfying or . Note that we use interchangeably with .
Take the lead Watson, and we will deduce the key properties of the arcsecant
function as we did above for arcsine and arctangent.
(a)
Using the definition of arcsecant given above, what are the domain and range of
the arcsecant function?
The domain of arcsecant is .
The range of arcsecant is .
(b)
Determine the following values exactly:
(c)
On the axes provided below, sketch a careful plot of the restricted secant
function on the intervals and along with its corresponding inverse, the
arcsecant function. Label at least three points on each curve so that each
point on the secant graph corresponds to a point on the arcsecant
graph.
(d)
True or false: ? truefalse Write a complete sentence to explain your reasoning.
Properties of the arcsecant function.
The restricted secant function, , is defined on the domain with range .
This function has an inverse function that we call the arcsecant function,
denoted .
The domain of is with range .
The arcsecant function is always increasing on each interval in its domain.
Recall that . Arcsecant and arccosine maintain a relationship as well,
though they are not reciprocals: For in the domain of arcsecant, .
Exploring Arcsecant
Sometimes we must rely on other properties of these functions and their relations to
more familiar functions to find solutions. In the following examples, we wish to find
in the range of arcsecant such that
(a)
We may use the relationship between arcsecant and arccosine to rewrite this
equation in terms of arccosine. In other words, since , for in the domain of
arcsecant,
(b)
Again, we use the relationship , for in the domain of arcsecant:
since the reciprocal of . Thus, we have
Let’s consider a couple more traditional problems combining secant and
arcsecant. Remember that we must be cautious of their respective domains and
ranges as with combinations of sine and arcsine and tangent and cotangent
explored above.
(c)
Recall from part (a), that we already solved the equation , and found that .
Hence, we can now plug that in to solve our current equation:
As we have observed previously with other trig inverses, we have for in the
domain of arcsecant. However, we must be careful in our application of this, as
exemplified by the next example.
(d)
Remember that we had to restrict the domain of secant in order to define the
inverse function, arcsecant. Arcsecant is thus the inverse of the restricted secant
function, which has domain . Observing that , and is therefore not in the
domain of the restricted secant function, we cannot simply treat arcsecant as
the inverse.
Instead, we begin by finding , which is equal to .
We now have , which we found to be equal to in part (a). Thus, we may
conclude that
We choose to define the restricted cosine, sine, tangent, and secant functions on
the respective domains , , , and . On each such interval, the restricted function
is strictly decreasing (cosine) or strictly increasing (sine, tangent, and secant),
and thus has an inverse function. The restricted sine and cosine functions each
have range , while the restricted tangent’s range is the set of all real numbers,
and the restricted secant’s range is . We thus define the inverse function of each
as follows:
i.
For any such that , the arccosine of (denoted ) is the angle in the
interval such that . That is, is the angle whose cosine is .
ii.
For any such that , the arcsine of (denoted ) is the angle in the
interval such that . That is, is the angle whose sine is .
iii.
For any real number , the arctangent of (denoted ) is the angle in
the interval such that . That is, is the angle whose tangent is .
iv.
For any real number , the arcsecant of (denoted ) is the angle in
the interval such that . That is, is the angle whose secant is .
The domain of is with corresponding range , and the arccosine function is
always decreasing. These facts correspond to the domain and range of the
restricted cosine function and the fact that the restricted cosine function is
decreasing on .
The domain of is with corresponding range , and the arcsine function is always
increasing. These facts correspond to the domain and range of the restricted
sine function and the fact that the restricted sine function is increasing on
.
The domain of is the set of all real numbers with corresponding range , and
the arctangent function is always increasing. These facts correspond
to the domain and range of the restricted tangent function and the
fact that the restricted tangent function is increasing on .
The domain of is ,1
with corresponding range . These facts correspond to the domain and range of
the restricted secant function.
