Inverse Trigonometric Functions

 

 

     When we introduced the graph of the sine function, we remarked that it repeats every 2π (this corresponds to a full rotation around the circle). Because of this property, the function y = sin(x) is not one-to-one. However, if we restrict the function to the interval   [-π/2, π/2], then it is one-to-one. See the figure below.

 

 

Figure 1: The unrestricted and restricted sine function

 

 

     Two notations are commonly used to denote the inverse sine function:

 

y = sin-1(x)     and     y = arcsin(x)

 

 

WARNING:

     y = sin-1(x) is not the same thing as .

 

     For example,  

 

 

 

     The graph of sin-1(x) can be found by reflecting the graph of the restricted sine function about the line y = x. Doing so, we have the following graph:

 

Figure 2: The graph of y = sin-1(x)

Example 1:

 

     Evaluate (i)  and (ii) .

 

Solution:

 

     (i)  is the number in the interval  whose sine is . Since

          , we conclude that .

     (ii)  is the number in the interval  whose sine is . Since

           , we conclude that .

 

 

     If f(x) and f  1(x) are any pair of inverse functions, then by definition,

 

f[f  1(x)] = x     for every x in the domain of f  1(x)

 

     and

 

f  1[f(x)] = x     for every x in the domain of f(x)

 

     Applying these facts to the restricted sine function and its inverse, we obtain the following two basic identities:

 

 

sin(sin-1(x)) = x     for every x in the interval [-1, 1]

sin-1(sin(x)) = x     for every x in the interval .

 

 

     The following example indicates that the domain restrictions accompanying these two identities cannot be ignored.

 

 

Example 2:

 

     Compute sin-1(sin(π)).

 

Solution:

 

     Notice that sin(π) = 0, so sin-1(sin(π)) = sin-1(0), but sin-1(0) = 0. Thus, we have that sin-1(sin(π)) = 0, not π. The reason why is because π is not in the domain of the restricted sine function.

     We can do the same thing for the cosine function. The graph of cosine repeats every 2π (this corresponds to a full rotation around the circle). Because of this property, the function y = cos(x) is also not one-to-one. However, if we restrict the function to the interval [0, π], then it is one-to-one. See the figure below.

 

 

Figure 3: The unrestricted and restricted cosine function

 

 

     Two notations are commonly used to denote the inverse cosine function:

 

y = cos-1(x)     and     y = arccos(x)

 

 

WARNING:

     y = cos-1(x) is not the same thing as .

 

     For example,  

 

 

 

     The graph of cos-1(x) can be found by reflecting the graph of the restricted cosine function about the line y = x. Doing so, we have the following graph:

 

Figure 4: The graph of y = cos-1(x)

 

 

     Again, we have two basic identities relating the function cos(x) and cos-1(x).

 

 

cos(cos-1(x)) = x     for every x in the interval [-1, 1]

cos-1(cos(x)) = x     for every x in the interval [0, π].

 

 

Example 3:

 

     Evaluate (i)  and (ii) .

 

Solution:

 

     (i)  is the number in the interval [0, π] whose cosine is 0. Since

          , we conclude that .

     (ii)  is the number in the interval [0, π] whose cosine is . Since

           , we conclude that .

 

 

     Just as there is a basic identity concerning sin(x) and cos(x), namely sin2(x) + cos2(x) = 1, there is also an identity concerning sin-1(x) and cos-1(x).

 

 

     for every x in the interval [-1, 1]

 

 

     Finally, we introduce the restricted tangent function and inverse tangent function. The graph of cosine repeats every π. Because of this property, the function y = tan(x) is also not one-to-one. However, if we restrict the function to the interval [-π/2, π/2], then it is one-to-one. See the figure below.

 

   

Figure 5: The unrestricted and restricted tangent function

 

 

 

 

     Two notations are commonly used to denote the inverse tangent function:

 

y = tan-1(x)     and     y = arctan(x)

 

 

WARNING:

     y = tan-1(x) is not the same thing as .

 

     For example,  

 

 

 

     The graph of tan-1(x) can be found by reflecting the graph of the restricted tangent function about the line y = x. Doing so, we have the following graph:

 

Figure 6: The graph of y = tan-1(x)

 

 

     Again, we have two basic identities relating the function tan(x) and tan-1(x).

 

 

tan(tan-1(x)) = x     for every real number x

tan-1(tan(x)) = x     for every x in the interval .

 

 

Example 4:

 

     Evaluate (i)  and (ii) .

 

Solution:

 

     (i)  is the number in the interval  whose tangent is -1. Since

          , we conclude that .

     (ii)  is the number in the interval  whose tangent is . Since

           , we conclude that .

 

 

Example 5:

 

     Simplify the quantity csc(tan-1(x)), where x > 0.

 

Solution:

 

     We let θ = tan-1(x). That is, we have that tan(θ) = x = x/1. Using this information, we can sketch a right triangle with an angle θ whose tangent is x. See Figure 7.

 

 

 

 

 

 

 

 


Figure 7: Graph of θ = tan-1(x)

 

 

     The Pythagorean Theorem tells us that the length of the hypotenuse in this triangle is equal to . Consequently, we have:

 

     .