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, Sin (x) , arcsine (x)
-1
•
•
•
•
•
Function is increasing
Takes on full range of values
Function is 1-1
Domain:  1, 1
Range:    ,  


2
2 
Evaluate: arcSin
3
2
• Asking the sine of what angle is
3
2
Find the following:
2
a) ArcSin
2
1
b) Sin (  )
-1
2
c) ArcSin

3
2
Inverse Cosine Function
• What can we restrict the domain of the cosine
curve to so that it is 1-1? 0 ,  
1
-1
Inverse Cosine, Cos (x) , arcCos (x)
-1
•
•
•
•
•
Function is increasing
Takes on full range of values
Function is 1-1
Domain:  1 , 1
Range:    ,  


2
2 
Evaluate: ArcCos (-1)
• The Cosine of what angle is -1?
Evaluate the following:
a) Cos -1 ( 3 )
2
b) ArcCos (c) Cos -1 (-
1
2
2
2
)
)
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:
– Cos
-1
(0.28)
Determine the missing Coordinate
Determine the missing Coordinate
Use an inverse trig function to write θ
as a function of x.
2x
θ
x+3
Find the exact value of the expression.
Sin [ ArcCos
 2
 
 3
]
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 first
2) Substitute in that value
3) Evaluate outermost function
Remember t
hat f (f (x) )  x and f
-1
As long as x is in the domain
-1
(f (x) )  x
of the necessasry
function
Sin (arcCos
1
2
)
Evaluate the innermost function first:
arcCos ½ =
Substitute that value in original problem
Sin

3
7 

Cos  Sin

6 

-1

Tan  Cos

-1
5 

13 
How do we evaluate this?
Let θ equal what is in parentheses
  Cos
-1
5
13
 Cos  
5
13
Cos  
5
13
13
θ
5
12

Tan  Cos

-1
5 

13 
 Tan 
How do we evaluate this?
Let θ equal what is in parentheses
Use the triangle to answer the question
Tan  
5
12
13
θ
5
12

Csc  Tan

-1
15 

8 
Sin Sin
-1
0.2 
What is different about this problem?
Is 0.2 in the domain of the arcSin?
Then Sin Sin
-1
0.2   0 . 2
Sin
-1
4 

 Sin

3 

What is different about this problem?
Is
4
in the domain
of the Sin function?
3
Since it is not, we must evaluate
Sin
4
3
Graph of the ArcSin
Y



X = Sin Y
1
2
 1
3
6
3

2
0

6

3

2
2
0
1
2
3
2
1
Graph of the ArcSin
Graph of ArcCos
Y
X = Sin Y
0
1

3
2
6

1
2
3

0
2
2
3
5
6

 1

2
3
2
1
Graph of the ArcCos