Inverse Trig Functions 6.1

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# Inverse Trig Functions 6.1 - PowerPoint PPT Presentation

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 Functions6.1

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

doesn’t exist.

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 Wave
• Domain/range of restricted wave?
• Domain/range of inverse?
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
• 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
• 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 Functions
• Find the value of:
• sin-1 2
• This means at what angle is the sin = 2 ?
• 2 falls outside the range of a sine wave and outside the domain of the inverse sine wave
Cosine Wave
• We must restrict the domain
• Now the inverse
Cosine Wave

doesn’t exist.

Tangent Wave
• We must restrict the domain
• Now the inverse
Graphing Utility: Graphs of Inverse Functions

–1.5

1.5

–

2

–1.5

1.5

–

–3

3

–

Graphing Utility:Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

Graphing Utility: Inverse Functions

Graphing Utility:Approximate the value of each expression.

a. cos–1 0.75

b. arcsin 0.19

c. arctan 1.32

d. arcsin 2.5

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
• Find the exact value of
• The sine angles must be in QI or QIV, so we must use the reference angle
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!
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
• Find the exact value of
• Cos is negative in QII and III, but the inverse is restricted to QII.

12

-7

You Do
• Find the exact value of