Inverse Trig FunctionsInverse Trig Functions6.16.1

Inverse Trig FunctionsInverse Trig Functions6.16.1

JMerrill, 2007JMerrill, 2007

Revised 2009Revised 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 ,

2 2

Sine Wave• Quadrant IV is • Quadrant I is • Answers must be in one of those

two quadrants or the answer doesn’t exist.

,02

0,2

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?

: ,2 2

: 1,1

D

R

: 1,1

: ,2 2

D

R

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

Cosine Wave• We must restrict the domain• Now the inverse

: 0,

: 1,1

D

R

: 1,1

: 0,

D

R

Cosine Wave• Quadrant I is • Quadrant II is • Answers must be in one of those

two quadrants or the answer doesn’t exist.

0,2

,2

Tangent Wave

Tangent Wave• We must restrict the domain• Now the inverse

Graphing Utility: Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

–1.5 1.5

–

–1.5 1.5

2

–

–3 3

–

Set calculator to radian mode.

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

Set calculator to radian mode.

Composition of Functions

• Find the exact value of•

• Where is the sine =• Replace the parenthesis in the

original problem with that answer• Now solve

1 2sin sin

2

22 4

sin4 2

2

Example• Find the exact value of

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

•

1 3sin sin

4

2sin

4 2

4

1 13sin sin sin sin

4 4

1 2sin

2

4

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!

1 1cos cos

2

12

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

2tan arccos

3

2

3 5

yt n

52

ax

Last Example• Find the exact value of• Cos is negative in QII and III, but

the inverse is restricted to QII.

1 7tan cos

12

-7

1295ytan

x957

You Do• Find the exact value of

1 3tan sin

7

3 1020