In this discussion, we are going to learn how to evaluate inverse tangent.
Key things to consider:
🔹Observing the domain
🔹Observing the Unit Circle
🔹Observing the Range
🔹Choosing The Value
Let us first look at a few concepts, then we will work our example together.
◻️◻️◻️Concepts◻️◻️◻️
When working with inverse trig functions, in particularly, inverse tangent, you will have an equation in the form
y = tan^-1(x).
🔷Observing the Domain
Note that the domain of tan^-1(x) is the set of all x such that tan^-1(x) is defined.
In other words, you can think of the domain as all your possible “input” values.
Where is tan^-1(x) defined?
It is defined on the interval:
(-inf, inf)
Thus, we say that the domain of tan^-1(x) is:
(-inf, inf)
We check to see if the x-value satisfies the domain.
If the x-value does not satisfy the domain, we simply say that there is no solution.
If the x-value is within the domain, we then take the tangent of both sides to get:
tan(y) =x
🔷Observing the Unit Circle
Looking at the unit circle, you will see terminal points in the form, (a,b).
Remember that tan(y) takes the value “b/a”.
We then search for all the values of y such that
tan(y) = x,
where x= b/a given terminal points (a,b).
Once we find all the values for y, we consider the range of tan^-1(x).
🔷Observing the Range
While the domain of tan^-1(x) is your possible “input” values, the range is your possible “output” values.
The range of tan^-1(x) is:
(-pi/2, pi/2)
which is the right region of the unit circle, but not including -pi/2 or pi/2.
Note that 3pi/2 (in the positive direction)
is the same as -pi/2 (in the negative direction).
If you would like to know more of how the domain and range of inverse tangent was discovered, go to the following article.
Graphing Inverse Tangent (ArcTangent) and Identifying the Domain and Range
🔷Choosing the Value
You may find that there are multiple values for y; however, we must choose the value for y that satisfies the range.
Now, let us look at our example.
◻️◻️◻️Example◻️◻️◻️
Let’s say we have,
y = tan^-1(-1)
Below are the steps:
🔷Observing the Domain
Note that the domain of tan^-1(x) is
(-inf, inf)
Since x is equal to -1 in our example, we know that it satisfies the domain.
Since our x-value is within the domain of tan^-1(x), we then take the tangent of both sides to get:
tan(y) = -1
🔷Observing the Unit Circle
Looking at the unit circle, we search for all the values of y such that tan(y) = -1.
Or sometimes I like to ask myself, “Where on the unit circle does tan(y)= -1?”
Remember, we want tan(y) = b/a = -1, given coordinates (a,b).
This means that we are looking in the second or fourth quadrant for coordinates:
(a,b) = (-sqrt(2)/2), sqrt(2)/2)
and
(a,b) = (sqrt(2)/2), -sqrt(2)/2)
because
b/a = – sqrt(2)/2 / sqrt(2)/2 = -1
and
b/a = sqrt(2)/2 / -sqrt(2)/2 = -1.
On the unit circle, we have terminal points:
(a,b) = (-sqrt(2)/2) , sqrt(2)/2) at the following angles:
3pi/4 (If we go in the positive direction)
-5pi/4 (If we go in the negative direction)
and we have terminal points
(a,b) = (sqrt(2)/2 , -sqrt(2)/2) at the following angles:
7pi/4 (If we go in the positive direction)
-pi/4 (If we go in the negative direction)
Thus, when
y= 3pi/4, -5pi/4, 7pi/4, and -pi/4
we get:
tan(3pi/4)= sqrt(2)/2 / sqrt(2)/2 = -1
tan(-5pi/4)= sqrt(2)/2 / sqrt(2)/2 = -1
tan(7pi/4)= -sqrt(2)/2 / sqrt(2)/2) = – 1
tan(-pi/4)= -sqrt(2)/2 / sqrt(2)/2) = – 1
🔷Observing the Range
Now, we cannot choose any value for y in our ultimate answer. We must consider the range of tan^-1(x).
The range of tan^-1(x) is
(-pi/2, pi/2)
which is the right region of the unit circle, but not including -pi/2 and pi/2.
🔷Choosing the Value
Remember that we must be sure that our value for y is within the range!
We see that the only value within the range is:
-pi/4, because 7pi/4 goes outside the range before it comes back around.
Thus, we choose -pi/4.
Hence,
tan(-pi/4) =-1
which means that
-pi/4 = tan^-1(-1),
and we say that
y= -pi/4
We are done!
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Need more examples?
Here are segregated blog posts from MathAngel369:
Inverse Sine | How to Find the Exact Value
Inverse Cosine | How to Find the Exact Value
Inverse Cotangent | How to Find the Exact Value
Inverse Secant | How to Find the Exact Value
Inverse Cosecant | How to Find the Exact Value
🔑 Practice is the key! If you would like more examples, practice problems with the answers, quizzes with the answers, and more regarding inverse trig functions, consider taking my Math Course on Inverse trig functions!