4.7 Inverse Trig Functions

Does the Sine function have an inverse?

1

-1

What could we restrict the domain to so that the sine function does have an inverse?

1

-1

2 ,

2

Inverse Sine, , arcsine (x)

• Function is increasing• Takes on full range of values• Function is 1-1• Domain: • Range:

(x)Sin -1

2 ,

2

1 1,

Evaluate: arcSin

• Asking the sine of what angle is

2

3

2

3

Find the following:

a) ArcSin

b)

c) ArcSin 2

3

2

2

)2

1(Sin 1-

Inverse Cosine Function

• What can we restrict the domain of the cosine curve to so that it is 1-1?

1

-1

, 0

Inverse Cosine, , arcCos (x)

• Function is increasing• Takes on full range of values• Function is 1-1• Domain: • Range:

(x)Cos-1

2 ,

2

1 , 1

Evaluate: ArcCos (-1)

• The Cosine of what angle is -1?

Evaluate the following:

a)

b) ArcCos

c)

)2

3(Cos 1-

)2

1(-

)2

2(-Cos 1-

ArcTan (x)

• Similar to the ArcSin (x)

• Domain of Tan Function:

• Range of Tan Function:

arcCos (0.28)

• Is the value 0.28 on either triangle or curve?

• Use your calculator:– (0.28)Cos-1

Determine the missing Coordinate

Determine the missing Coordinate

Use an inverse trig function to write θ as a function of x.

θ

2x

x + 3

Find the exact value of the expression.

Sin [ ArcCos ]

3

2

4.7 Inverse Trig Functions

So far we have:

1) Restricted the domain of trig functions to find their inverse

2) Evaluated inverse trig functions for exact values

3) Found missing coordinates on the graphs of inverses

4) Found the exact values of compositions

Composition of Functions

1) Evaluate innermost function first2) Substitute in that value3) Evaluate outermost function

x ) (x) (f f and x ) (x)(f fhat Remember t -1-1 function necessasry theofdomain in the is x as long As

Sin (arcCos )2

1

Evaluate the innermost function first:arcCos ½ =

Substitute that value in original problem

3Sin

6

7Sin Cos 1-

13

5 CosTan 1-

How do we evaluate this?

Let θ equal what is in parentheses

13

5Cos 1-

13

5 Cos

13

5Cos

θ5

13 12

13

5 CosTan 1-

How do we evaluate this?

Let θ equal what is in parentheses

Use the triangle to answer the question

Tan

θ5

13 12

12

5Tan

8

15- TanCsc 1-

0.2 SinSin -1

What is different about this problem?

Is 0.2 in the domain of the arcSin?

2.00.2 SinSinThen -1

3

4Sin Sin 1-

What is different about this problem?

3

4Sin evaluatemust wenot, isit Since

function?Sin theofdomain in the 3

4 Is

Graph of the ArcSinY X = Sin Y

2

3

6

0 0

6

3

2 1

23

21

23

21

1

Graph of the ArcSin

Graph of ArcCosY X = Sin Y

32

6

5

0

6

3

2 0

12

3

21

23

21

1

Graph of the ArcCos