4.7 Inverse Trig Functions

Inverse trig functions

• What trig functions can we evaluate without using a calculator?

– Sin

– Cos

– Tan

– Sin

Inverse Trig Functions

• What does an inverse function do?– Finds the input of a function when given the output

• How can we determine if a function has an inverse?– Horizontal Line Test– If any horizontal line intersects the graph of a function

in more than one point, the function does not have an inverse

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-

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

ArcTan (x)

• Similar to the ArcSin (x)

• Domain of Tan Function:

• Range of Tan Function:

Composition of Functions

From Algebra II:

If two functions, f(x) and (x), are inverses, then their compositions are:

f((x)) = x and (f(x)) = x

Inverse Properties of Trig Functions

• If -1 ≤ x ≤ 1 and - ≤ y ≤ , thenSin (arcSin x) = x and arcSin (Sin y) = y

• If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π, thenCos (arcCos x) = x and arcCos (Cos y) = y

• If x is a real number and - < y < , thenTan (arcTan x) = x and arcTan (Tan y) = y

Inverse Trig Functions

• Use the properties to evaluate the following expression:

Sin (ArcSin 0.3)

Inverse Trig Functions

• Use the properties to evaluate the following expression:

ArcCos (Cos )

Inverse Trig Functions

• Use the properties to evaluate the following expression:

ArcSin (Sin 3π)

Inverse Trig Functions

• Use the properties to evaluate the following expression:

a) Tan (ArcTan 25)

b) Cos (ArcCos -0.2)

c) ArcCos (Cos )

4.7 Inverse Trig Functions

Inverse Trig Functions

• Yesterday, we only had compositions of functions that were inverses

• When we have a composition of two functions that are not inverses, we cannot use the properties

• In these cases, we will draw a triangle

Inverse Trig Functions

• Sin (arcTan )– Let u = whatever is in parentheses

• u = arcTan → Tan u =

Inverse Trig Functions

• Sec (arcSin )

Inverse Trig Functions

• Sec (arcSin )

• Cot (arcTan - )

• Sin (arcTan x)

Inverse Trig Functions

• In this section, we have:– Defined our inverse trig functions for specific dom

ains and ranges– Evaluated inverse trig functions– Evaluated compositions of trig functions

• 2 Functions that are inverses• 2 Functions that are not inverses by evaluating the inne

r most function first• 2 Functions that are not inverses by drawing a triangle

Sine Function

-

1

-1

Cosine Function

π

1

-1

Tangent Function

-

Evaluating Inverse Trig Functions

a) arcTan (- )

b) )

c) arcSin (-1)

Composition of Functions

• When the two functions are inverses:

a) Sin (arcSin -0.35)

b) arcCos (Cos )

Composition of Functions

• When the two functions are not inverses:

a) (Cos )

b) arcTan (Sin )

Composition of Functions

• When the two functions are not inverses:

a) Sin (arcCos )

b) Cot ( )

4.7 Inverse Trig Functions

Inverse Trig Functions

• Evaluate the following function:f(x) = Sin (arcTan 2x)

In your graphing calculator, graph both of these functions.

Inverse Trig Functions

• Solve the following equation for the missing piece:

arcTan = arcSin (___)

Inverse Trig Functions• Find the missing pieces in the following

equations:

a) arcSin = arcCos (___)

b) arcCos = arcSin (___)

c) arcCos = arcTan (___)

Inverse Trig Functions

Inverse Trig Functions

Composition of Functions

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

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

5

12Tan

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