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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
ad