Math 1330
Section 5.4
Section 5.4: Inverse Trigonometric Functions
Here is the graph of f (x )  sin( x ) on the interval [2 ,2 ] . Sin(x) is not a one-to one function.
Restricted Sine Function
y
x
To get around this, we begin to limit the domain of each of the trig functions, so that over the
restricted domain, the function is one-to-one.
“Restricted” sine function:
Inverse sine function:
f ( x)  sin 1 x or
f ( x)  arcsin x
1


2

-1
2
Domain: ________________
Domain: _________________
Range: _ _________________
Range: __________________
sin  sin 1  x    x when x is in the interval ________________.
sin 1  sin  x    x when x is in the interval ________________.
Math 1330
Section 5.4
We have the same situation with cosine and tangent. We must restrict the domains in order to get
inverses.
Restricted”cosine function:
Inverse cosine function:
f ( x)  cos 1 x or
f ( x)  arccos x
1

-1
Domain:_________________
Domain:_________________
Range:__________________
Range:__________________
cos  cos1  x    x when x is in the interval ________________.
cos1  cos  x    x when x is in the interval ________________.
Example 1:

3
Evaluate cos 1  
 .
 2 
Math 1330
Section 5.4
“Restricted” tangent function:
Inverse tangent function:
f ( x)  tan 1 x or
f ( x)  arctan x
1


2

-1
2
Domain:_________________
Domain:_________________
Range:__________________
Range:__________________
tan  tan 1  x    x when x is in the interval ________________.
tan 1  tan  x    x when x is in the interval ________________.
Example 2: Evaluate arctan(1) .
 2
Example 3: Evaluate sin 1 
 .
2


Math 1330
Section 5.4
Here is a summary of properties that maybe helpful when evaluating inverse trig function:
sin(sin 1 (x ))  x
1
cos(cos (x ))  x
tan(tan 1 (x ))  x
on  1,1
on  1,1
on   , 
cos 1 (cos( x ))  x
 
on  , 
 2 2 
on 0,  
tan 1 (tan(x ))  x
on
sin 1 (sin( x ))  x
  , 


 2 2
Notation: sin 1 (x )  arcsin( x ) . This is true for all trig functions. The arc in front of the trig
function is the older notation but you will see both notations and need to know their meanings.

3
Example 4: Evaluate sin 1  
 .
2


Example 5: Evaluate tan 1  0  .

2
Example 6: Evaluate arccos  
 .
2


 1
Example 7: Evaluate sin 1    .
 2
Math 1330
Section 5.4
 1 
Example 8: Evaluate tan 1  
.
3

Example 9: Evaluate sin 1  .
Example 10: Evaluate cosarccos( 2) .

 4 
Example 11: Evaluate sin  sin 1   
 7 

1 

Example 12: Evaluate tan sin 1    .
 2 


 2 
Example 13: Evaluate cos  sin 1    .
 7 
