Objective:
 Determine the values of the inverse sine, cosine, and tangent functions so that we can solve trig equations.
Find the inverse trig buttons on your calculator and evaluate the following: sin

1
1
 
First, some background….. cos

1 

1
4

 tan

1
1
 
Suppose
( ) and
( ) are inverse functions. The inverse of f is denoted f

1
( )
, and the inverse of g is denoted g

1
( )
.
Inverse SIN
However, with trig functions, the Horizontal Line Test would tell us that the inverse of is not itself a function. So how do we handle that? Let’s look at the graph.

Example 1
Evaluate the following (Think about where your answers HAVE to come from???!!!)
Inverse
The tangent sin cos

1

1
 
 

 sin

1  
1
2
 
Inverse COS
Notice how, with the cosine function, if we tried to take the same piece as we did with the inverse sine function, the HLT would still fail. So we change our interval to
__________
Example 2 cos

1


2
2



TAN function shows

some similarity to the sine function, in that the interval to be used for the inverse TAN can be __________.
Example 3 tan

1
   tan

1
 

Example 4 – Free for All sin

1

 
2
3

  cos

1
  tan

1


3
3


 sin

1

 
2
2

  cos

1
 
0.5
 
Create a flowchart…..
To illustrate the process one uses to evaluate an inverse trig function.

WARMUP – Day 2
I know this is challenging, and that you’ve never seen it – but if you remember
PEMDAS, you might surprise yourself!
 tan sin

1



1
2





Quick Composition Review
You remember
( ( ))
??? Let’s suppose that
( )

3 x

1
.
What is the inverse of
( )

3 x

1
?
We learned yesterday that the inverse is written as f

1
( )
. What is
(

1
( ))
?
So……if you had to try and generalize, evaluate the following:
 sin(sin ( ))

1

 
1

1 

Example 5 – Mind Blowing

1
 sin sin


5

4 

1
 cos cos


7

6 

1

1 



1
 cos sin


7

6

 sin cos


1


2
2



 




1

 
2
3



 
Ranges of the Inverse Functions
Function

1

1

1

1

1

1
 tan cos

1
1
 


Range
 
  
2 y
2
0
  

2 y

2
0
  
0

2 y

, y


2

2
, y

0
Example 6 – The Charts Return
 tan cos

1



1
3





 cos sin

1



1
3







1 csc ( 2)

1 cot (1)
