L26 The Inverse Trigonometric Functions by fad10689

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									L26       The Inverse Trigonometric Functions

The six trigonometric functions are not one-to-one on
their domains.

To define the inverse of a trigonometric function, we
consider the function on the restricted domain where it is
one-to-one. The restricted domain we will call the
Interval of Definition.

Intervals of Definition and Inverses of the Trigonometric
Functions:

f ( x ) = sin x        Interval of Definition:




f −1 ( x ) = sin −1 x :   Domain:            Range:
Note: f −1 ( x ) = sin −1 x is an odd function.

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f ( x ) = cos x           Interval of Definition:




f −1 ( x ) = cos −1 x :   Domain:           Range:

f ( x ) = tan x           Interval of Definition:




f −1 ( x ) = tan −1 x : Domain:           Range:
Note: f −1 ( x ) = tan −1 x is an odd function.

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f ( x ) = cot x           Interval of Definition:




f −1 ( x ) = cot −1 x :   Domain:           Range:

Notes:
The domains of the inverse trigonometric functions are
the ranges of the corresponding trigonometric function.

The ranges of the inverse trigonometric functions are the
Intervals of Definitions of the corresponding
trigonometric functions.

Important: Values of an inverse trigonometric function
are the angles from the Interval of Definition (see the
ranges of the inverse functions).


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Cancellation Rules (Part I)

         Identities:
       sin ( sin −1 x ) = x            −1 ≤ x ≤ 1
       cos ( cos −1 x ) = x            −1 ≤ x ≤ 1
       tan ( tan −1 x ) = x           −∞ < x < +∞
       cot ( cot −1 x ) = x           −∞ < x < +∞

Example: Evaluate:
   sin ( sin −1 (1/ 4) )

         ⎛       3⎞
     cos ⎜ cos −1 ⎟
         ⎝       2⎠


Definitions of the inverse trigonometric functions come
from the Cancellation Rules (Part I):

The inverse sine, cosine, tangent or cotangent of a real
number x is the angle from the Interval of Definition
whose sine, cosine, tangent or cotangent is equal to x.

Note: Number x must be in the domain of the inverse
function.

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Sometimes, one can easily find the exact value of an
inverse function by using the definitions above. In some
complicated cases, however, it is useful to know the
extended procedure of

Finding Exact Value of the Inverse Trigonometric
Function

Example: Find sin −1 x     ( −1 ≤ x ≤ 1).

1. Denote:               sin −1 x = θ

2. Compose sine function to both sides to cancel the
inverse:
                       x = sin θ

3. Pick up the appropriate value for θ from the Interval
of Definition.




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Example: Find the exact values (without a calculator)

sin −1 1 =


       ⎛ 1⎞
cos −1 ⎜ − ⎟ =
       ⎝ 2⎠

        ⎛ 3⎞
arcsin ⎜ − ⎟ =                                         cos −1 1 =
        ⎝ 2 ⎠
tan −1 (−1) =                                                  (
                                                       cot −1 − 3 =        )
             π       π       π           π                 π          π          π      π
  x     −        −       −           −             0
             2     3       4        6                      6          4          3      2
                    3       2       1                      1           2          3
sin x    −1      −       −        −                0                                    1
                   2       2        2                      2          2          2
                                    1                      1
tan x ----       − 3     −1      −                 0                  1           3    ----
                                     3                      3

                 π       π       π             π        2π          3π          5π
  x          0
                  6      4       3             2         3           4           6
                                                                                       π
                   3      2      1                        1            2           3
cos x        1                                 0        −          −           −       −1
                  2      2       2                        2           2           2
                                 1                        1
cot x ----           3   1                     0       −            −1         − 3     ----
                                  3                        3

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The Inverse Secant Function
    f ( x) = sec x    Interval of Definition:




f −1 ( x) = sec−1 x




Domain:

Range:

HA:

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The Inverse Cosecant Function
f ( x) = csc x        Interval of Definition:




f −1 ( x) = csc−1 x




Domain:

Range:

HA:

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Note: The functions sec x and csc x have the same
Intervals of Definition as cos x and sin x , respectively,
except for the points where secant and cosecant are
undefined.

Example: Find the exact values without a calculator.

csc −1 2 =

sec−1 1 =


       ⎛ 2 ⎞
csc −1 ⎜ −  ⎟
       ⎝   3⎠




sec −1 ( −2 )




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Cancellation Rules (Part II)

          Identities:                  (Interval of Definition)
                                                 ⎡ π π⎤
       sin −1 ( sin θ ) = θ                 θ ∈ ⎢− , ⎥
                                                 ⎣ 2 2⎦
                                                ⎛ π π⎞
       tan −1 ( tan θ ) = θ                 θ ∈⎜ − , ⎟
                                                ⎝ 2 2⎠
       cos −1 ( cosθ ) = θ                    θ ∈ [ 0, π ]
       cot −1 ( cot θ ) = θ                   θ ∈ ( 0, π )

Note: If θ is not in the Interval of Definition, the
Identities do not hold.

In order to extend the use of the cancellation rules for any
θ in the Domain of the trigonometric function, we must
replace angle θ with the unique angle θ1 from the
Interval of Definition such that
                       sin θ1 = sin θ .

The angle θ1 can be found by using Periodic Properties,
Even-Odd Properties, graphs and/or the Unit Circle.

The modified Cancellation Rule is
             sin −1 (sin θ ) = sin −1 (sin θ1 ) = θ1


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Example: Evaluate:
       ⎛    π⎞
cos −1 ⎜ cos ⎟
       ⎝    5⎠

sin −1 ( sin(14π / 5) )




cos −1 ( cos(6π / 5) )




       ⎛     27π ⎞
tan −1 ⎜ tan     ⎟
       ⎝      4 ⎠




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Example: Give each value without using a calculator.

        ⎛        ⎛ 1 ⎞⎞
    sin ⎜ cos −1 ⎜ − ⎟ ⎟
        ⎝        ⎝ 4 ⎠⎠




    Example: Write as an expression in u.

    sin ( tan −1 u ) =

    cos ( tan −1 u ) =




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