Inverse Trig Functions 6.1 - PowerPoint PPT Presentation

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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

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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

• Quadrant IV is

• Quadrant I is

• Answers must be in one of those two quadrants or

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 ?

• 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 Wave

• We must restrict the domain

• Now the inverse

Cosine Wave

• Quadrant I is

• Quadrant II is

• Answers must be in one of those two quadrants or

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.

Set calculator to radian mode.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

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

• 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