4_7 INVERSE TRIG FNS.pdf

  • View
    213

  • Download
    0

Embed Size (px)

Transcript

  • 4.7 INVERSE TRIGONOMETRIC FUNCTIONS

    Copyright Cengage Learning. All rights reserved.

  • 2

    Evaluate and graph the inverse sine function.

    Evaluate and graph the other inverse trigonometric functions.

    Evaluate and graph the compositions of trigonometric functions.

    What You Should Learn

  • 3

    Inverse Sine Function

  • 4

    Inverse Sine Function

    For a function to have an inverse function, it must be one-to-onethat is, it must pass the Horizontal Line Test.

    sin x has an inverse function

    on this interval.

    Restrict the domain to the interval / 2 x

    / 2, the following properties hold.

    1. On the interval [ / 2, / 2], the function

    y = sin x is

    increasing.

    2. On the interval [ / 2, / 2], y = sin x

    takes on its full

    range of values, 1 sin x 1.

    3. On the interval [ / 2, / 2], y = sin x is

    one-to-one.

    By definition, the values of inverse

    trigonometric functions are always

    in radians.

  • 5

    Inverse Sine Function

    On the restricted domain / 2 x / 2, y = sin x has a unique inverse function called the inverse sine function.

    y = arcsin x or y = sin 1 x.

    means the angle (or arc) whose sine is x.

  • 6

    Inverse Sine Function

  • 7

    Example 1 Evaluating the Inverse Sine Function

    If possible, find the exact value.

    a. b. c.

    Solution:

    a. Because for ,

    . Angle whose sine is

  • 8

    Example 1 Solution

    b. Because for ,

    .

    c. It is not possible to evaluate y = sin 1 x when x = 2

    because there is no angle whose sine is 2.

    Remember that the domain of the inverse sine function

    is [1, 1].

    contd

    Angle whose sine is

  • 9

    Example 2 Graphing the Arcsine Function

    Sketch a graph of y = arcsin x.

    Solution:

    In the interval [ / 2, / 2],

  • 10

    Example 2 Solution

    y = arcsin x

    Domain: [1, 1]

    Range: [ / 2, / 2]

    contd

  • 11

    Other Inverse Trigonometric

    Functions

  • 12

    Other Inverse Trigonometric Functions

    The cosine function is decreasing and one-to-one on the

    interval 0 x .

    On the interval 0 x the cosine function has an inverse functionthe inverse cosine function:

    y = arccos x or y = cos 1 x.

    cos x has an inverse function on this interval.

  • 13

    Other Inverse Trigonometric Functions

    DOMAIN: [1,1] RANGE:

    DOMAIN: [1,1]

    RANGE: [0, ] DOMAIN:

    RANGE:

  • 14

    Example 3 Evaluating Inverse Trigonometric Functions

    Find the exact value.

    a. arccos b. cos 1(1)

    c. arctan 0 d. tan 1 (1)

    Solution:

    a. Because cos ( / 4) = , and / 4 lies in [0, ],

    . Angle whose cosine is

  • 15

    Example 3 Solution

    b. Because cos = 1, and lies in [0, ],

    cos 1(1) = .

    c. Because tan 0 = 0, and 0 lies in ( / 2, / 2),

    arctan 0 = 0.

    Angle whose cosine is 1

    Angle whose tangent is 0

    contd

  • 16

    Example 3 Solution

    d. Because tan( / 4) = 1, and / 4 lies in ( / 2, / 2),

    tan 1 (1) = . Angle whose tangent is 1

    contd

  • 17

    Compositions of Functions

  • 18

    Compositions of Functions

    For all x in the domains of f and f 1, inverse functions have the

    properties

    f (f 1(x)) = x and f 1 (f (x)) = x.

  • 19

    Compositions of Functions

    These inverse properties do not apply for arbitrary values

    of x and y.

    .

    The property arcsin(sin y) = y

    is not valid for values of y outside the interval [ / 2, / 2].

  • 20

    Example 5 Using Inverse Properties

    If possible, find the exact value.

    a. tan[arctan(5)] b. c. cos(cos 1 )

    Solution:

    a. Because 5 lies in the domain of the arctan function, the inverse property applies, and you have

    tan[arctan(5)] = 5.

  • 21

    Example 5 Solution

    b. In this case, 5 / 3 does not lie within the range of the arcsine function, / 2 y / 2.

    However, 5 / 3 is coterminal with

    which does lie in the range of the arcsine function,

    and you have

    contd

  • 22

    Example 5 Solution

    c. The expression cos(cos 1) is not defined because cos 1 is not defined.

    Remember that the domain of the inverse cosine

    function is [1, 1].

    contd

  • 23

    Find the exact value: tan arccos2

    3

    Let = arccos2

    3, then cos =

    2

    3> 0, therefore, u is in QI

    tan arccos2

    3= tan =

    sin

    cos =

    1 23

    2

    23

    =5

    2

    Example 6 Evaluating Compositions of Functions

  • 24

    Find the exact value: cos arc(3

    5)

    Let = arc(3

    5), then sin =

    3

    5< 0, therefore, u is in

    Q IV

    cos arc(3

    5) = 1

    3

    5

    2

    =4

    5

    Example 6 Evaluating Compositions of Functions

  • 25

    Example

    Write the expression as an algebraic expression in x:

    sin arccos 3 , 0 1

    3

    sin arccos 3 = 1 92