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Evaluating compositions of the form f ( f −1 ( y )) and f −1 ( f ( x ))

For any trigonometric function, f ( f 1 ( y ) ) = y for all y in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of f was defined to be identical to the domain of f 1 . However, we have to be a little more careful with expressions of the form f 1 ( f ( x ) ) .

Compositions of a trigonometric function and its inverse

sin ( sin 1 x ) = x for 1 x 1 cos ( cos 1 x ) = x for 1 x 1 tan ( tan 1 x ) = x for < x <


sin 1 ( sin x ) = x only for  π 2 x π 2 cos 1 ( cos x ) = x only for  0 x π tan 1 ( tan x ) = x only for  π 2 < x < π 2

Is it correct that sin 1 ( sin x ) = x ?

No. This equation is correct if x belongs to the restricted domain [ π 2 , π 2 ] , but sine is defined for all real input values, and for x outside the restricted interval, the equation is not correct because its inverse always returns a value in [ π 2 , π 2 ] . The situation is similar for cosine and tangent and their inverses. For example, sin 1 ( sin ( 3 π 4 ) ) = π 4 .

Given an expression of the form f −1 (f(θ)) where f ( θ ) = sin θ ,   cos θ ,  or  tan θ , evaluate.

  1. If θ is in the restricted domain of f ,  then  f 1 ( f ( θ ) ) = θ .
  2. If not, then find an angle ϕ within the restricted domain of f such that f ( ϕ ) = f ( θ ) . Then f 1 ( f ( θ ) ) = ϕ .

Using inverse trigonometric functions

Evaluate the following:

  1. sin 1 ( sin ( π 3 ) )
  2. sin 1 ( sin ( 2 π 3 ) )
  3. cos 1 ( cos ( 2 π 3 ) )
  4. cos 1 ( cos ( π 3 ) )
  1. π 3  is in  [ π 2 , π 2 ] , so sin 1 ( sin ( π 3 ) ) = π 3 .
  2. 2 π 3  is not in  [ π 2 , π 2 ] , but sin ( 2 π 3 ) = sin ( π 3 ) , so sin 1 ( sin ( 2 π 3 ) ) = π 3 .
  3. 2 π 3  is in  [ 0 , π ] , so cos 1 ( cos ( 2 π 3 ) ) = 2 π 3 .
  4. π 3  is not in  [ 0 , π ] , but cos ( π 3 ) = cos ( π 3 ) because cosine is an even function.
  5. π 3  is in  [ 0 , π ] , so cos 1 ( cos ( π 3 ) ) = π 3 .
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Evaluate tan 1 ( tan ( π 8 ) ) and tan 1 ( tan ( 11 π 9 ) ) .

π 8 ; 2 π 9

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Evaluating compositions of the form f −1 ( g ( x ))

Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form f 1 ( g ( x ) ) . For special values of x , we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is θ , making the other π 2 θ . Consider the sine and cosine of each angle of the right triangle in [link] .

An illustration of a right triangle with angles theta and pi/2 - theta. Opposite the angle theta and adjacent the angle pi/2-theta is the side a. Adjacent the angle theta and opposite the angle pi/2 - theta is the side b. The hypoteneuse is labeled c.
Right triangle illustrating the cofunction relationships

Because cos θ = b c = sin ( π 2 θ ) , we have sin 1 ( cos θ ) = π 2 θ if 0 θ π . If θ is not in this domain, then we need to find another angle that has the same cosine as θ and does belong to the restricted domain; we then subtract this angle from π 2 . Similarly, sin θ = a c = cos ( π 2 θ ) , so cos 1 ( sin θ ) = π 2 θ if π 2 θ π 2 . These are just the function-cofunction relationships presented in another way.

Given functions of the form sin 1 ( cos x ) and cos 1 ( sin x ) , evaluate them.

  1. If x  is in  [ 0 , π ] , then sin 1 ( cos x ) = π 2 x .
  2. If x  is not in  [ 0 , π ] , then find another angle y  in  [ 0 , π ] such that cos y = cos x .
    sin 1 ( cos x ) = π 2 y
  3. If x  is in  [ π 2 , π 2 ] , then cos 1 ( sin x ) = π 2 x .
  4. If x  is not in [ π 2 , π 2 ] , then find another angle y  in  [ π 2 , π 2 ] such that sin y = sin x .
    cos 1 ( sin x ) = π 2 y

Questions & Answers

what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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