In this discussion, we learn how to evaluate inverse cosecant.
Key things to consider:
🔹Observing the domain
🔹Converting csc(y) to sin(y)
🔹Observing the Unit Circle
🔹Observing the Range
🔹Choosing the Value
◻️◻️◻️Concepts◻️◻️◻️
When working with inverse trig functions, in particularly, inverse cosecant, the equation is given in the form
y = csc^-1(x).
🔷Observe the Domain
Now the domain of csc^-1(x) is the set of all x such that csc^-1(x) is defined.
In other words, the domain as all possible “input” values.
Note that the domain of csc^-1(x) is
(-inf,-1] U [1,inf)
Thus, check to see if the x-value satisfies the domain.
If the x-value does not satisfy the domain, the there is no solution.
If the x-value is within the domain, we then take the cosecant of both sides to get
csc(y) =x
🔷Convert csc(y) to sin(y)
Now it is much easier to evaluate sin(y) instead of csc(y).
Thus, use the fact that
csc(y) =1/sin(y) or sin(y) =1/csc(y)
Hence, csc(y) =x can be rewritten as
sin(y) = 1/x
🔷Observing the Unit Circle
Looking at the unit circle, there are coordinates in the form, (a,b).
Remember that sine take takes the “b” coordinate.
Search for all the values of y such that
sin(y) = 1/x,
where 1/x is in the “b” location.
Next, consider the range of csc^-1(x).
🔷Observe the Range
While the domain is all possible input values, the range is all possible output values.
The range of csc^-1(x) is
[-pi/2,0) U (0,pi/2],
which is the right region of the unit circle, but not including 0.
If you would like to know more of how the domain and range of inverse cosecant was discovered, go to this article.
🔷Choose the Value
You may find multiple values for y; however, always choose the value for y that satisfies the range.
◻️◻️◻️Example ◻️◻️◻️
Let us say you are given,
y = csc^-1(-2).
Here are the steps:
🔷Observe the Domain
Note that the domain of csc^-1(x) is
(-inf,-1] U [1,inf).
Since x=-2, the domain is satisfied.
Since the x-value is within the domain of csc^-1(x), take the cosecant of both sides to get
csc(y) = -2.
🔷Convert csc(y) to sin(y)
Again, it is much easier to evaluate sin(y) instead of csc(y).
So, use the fact that
csc(y) =1/sin(y) or sin(y) =1/csc(y)
Thus, csc(y) = -2 can be rewritten as
sin(y) = -1/2
🔷Observe the Unit Circle
Looking at the unit circle, search for all the values of y such that sin(y) = -1/2.
I like to ask myself, “Where on the unit circle does sin(y) = -1/2?
On the unit circle, there are coordinates
(a,b) = (sqrt(3)/2, -1/2) at the following angles:
11pi/6 (If we go in the positive direction)
-pi/6 (If we go in the negative direction)
and we have coordinates
(a,b) = (-sqrt(3)/2, -1/2) at the following angles:
7pi/6 (If we go in the positive direction)
-5pi/6 (If we go in the negative direction)
Remember, we observe the “b” coordinate when working with sine.
Thus, when:
y = 11pi/6, -pi/6, 7pi/6, -5pi/6,
we get:
sin(11pi/6)=-1/2
sin(-pi/6)=-1/2
sin(7pi/6)=-1/2
sin(-5pi/6)=-1/2
🔷Observe the Range
Now, we cannot choose any value for y in the final answer.
We must consider the range of ccs^-1(x).
The range of csc^-1(x) is
[-pi/2 , 0) U (0 , pi/2],
which is the right region of the unit circle, but not including zero.
🔷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/6.
Thus, we choose -pi/6.
Hence,
sin(-pi/6) =-1/2,
which means that
csc(-pi/6)=-2,
which means that
-pi/6 = csc^-1(-2),
and we say that
y=-pi/6.
We are done!
📝Suggested Blog Post
Inverse Sine | How to Find the Exact Value of Inverse Trig Functions
🔑 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!