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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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