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Inverse Trig Functions 6.1. JMerrill, 2007 Revised 2009. Recall. From College Algebra, we know that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test. Sine Wave.

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Inverse trig functions 6 1

Inverse Trig Functions6.1

JMerrill, 2007

Revised 2009


Recall
Recall

  • From College Algebra, we know that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test.


Sine wave
Sine Wave

  • From looking at a sine wave, it is obvious that it does not pass the Horizontal Line Test.


Sine wave1
Sine Wave

  • In order to pass the Horizontal Line Test (so that sin x has an inverse that is a function), we must restrict the domain.

  • We restrict it

    to


Sine wave2
Sine Wave

  • Quadrant IV is

  • Quadrant I is

  • Answers must be in one of those two quadrants or

    the answer

    doesn’t exist.


Sine wave3
Sine Wave

  • How do we draw inverse functions?

  • Switch the x’s and y’s!

Switching the x’s and y’s also means switching the axis!


Sine wave4
Sine Wave

  • Domain/range of restricted wave?

  • Domain/range of inverse?


Inverse notation
Inverse Notation

  • y = arcsin x or y = sin-1 x

  • Both mean the same thing. They mean that you’re looking for the angle (y)where sin y = x.


Evaluating inverse functions
Evaluating Inverse Functions

  • Find the exact value of:

  • Arcsin ½

    • This means at what angle is the sin = ½ ?

    • π/6

    • 5π/6 has the same answer, but falls in QIII, so it is not correct.


Calculator
Calculator

  • When looking for an inverse answer on the calculator, use the 2nd key first, then hit sin, cos, or tan.

  • When looking for an angle always hit the 2nd key first.

  • Last example: Degree mode, 2nd, sin, .5 = 30.


Evaluating inverse functions1
Evaluating Inverse Functions

  • Find the value of:

  • sin-1 2

    • This means at what angle is the sin = 2 ?

    • What does your calculator read? Why?

    • 2 falls outside the range of a sine wave and outside the domain of the inverse sine wave



Cosine wave1
Cosine Wave

  • We must restrict the domain

  • Now the inverse


Cosine wave2
Cosine Wave

  • Quadrant I is

  • Quadrant II is

  • Answers must be in one of those two quadrants or

    the answer

    doesn’t exist.



Tangent wave1
Tangent Wave

  • We must restrict the domain

  • Now the inverse


Graphing utility graphs of inverse functions
Graphing Utility: Graphs of Inverse Functions

–1.5

1.5

–

2

–1.5

1.5

–

–3

3

–

Graphing Utility:Graph the following inverse functions.

Set calculator to radian mode.

a. y = arcsin x

b. y = arccos x

c. y = arctan x


Graphing utility inverse functions
Graphing Utility: Inverse Functions

Graphing Utility:Approximate the value of each expression.

Set calculator to radian mode.

a. cos–1 0.75

b. arcsin 0.19

c. arctan 1.32

d. arcsin 2.5


Composition of functions
Composition of Functions

  • Find the exact value of

  • Where is the sine =

  • Replace the parenthesis in the original problem with that answer

  • Now solve


Example
Example

  • Find the exact value of

  • The sine angles must be in QI or QIV, so we must use the reference angle


Example1
Example

  • Find tan(arctan(-5))

    -5

  • Find

  • If the words are the same and the inverse function is inside the parenthesis, the answer is already given!


Example2
Example

  • Find the exact value of

  • Steps:

  • Draw a triangle using only the info inside the parentheses.

  • Now use your x, y, r’s to answer the outside term

3

2


Last example
Last Example

  • Find the exact value of

  • Cos is negative in QII and III, but the inverse is restricted to QII.

12

-7


You do
You Do

  • Find the exact value of


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