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Section 4.7
Inverse Trigonometric Functions
A brief review…..
1. If a function is one-to-one, the function has
an inverse that is a function.
2. If the graph of a function passes the
horizontal line test, then the function is oneto-one.
3. Some functions can be made to pass the
horizontal line test by restricting their
domains.
More…
4. If (a,b) is a point on the graph of f, then (b,a)
is a point on the graph of f-inverse.
5. The domain of f-inverse is the range of f.
6. The range of f-inverse is the domain of f.
7. The graph of f-inverse is a reflection of the
graph of f about the line y = x.
fx = sinx
y = sin x is graphed below. The restricted
portion is highlighted.
6
4
2
-π/2
-10
π/2
-5
-2
-4
-6
-8
5
10
The inverse sine function
• written y = sin−1x or y = arcsin x
• The domain of y = sin x is restricted to


2
x

2
• y = sin-1 x means that sin y = x (inverse x & y swapped)
• sin-1 x is the angle, between –π/2 and π/2,
(inclusive), whose sine value is x.
Find the exact value (in radians) of each of the following:
   
1
,
sin    Think: the angle in  2 2  whose sine is ½.
2
The answer is π6 because it is the angle in
1
   
 , 
 2 2
 1  11π
sin    
 2
6
1

2
sin  
 
 2 
1
whose sine is ½.
   
not in  , 
 2 2
4.7 – Inverse Trig Functions
y = cos x
[0,π]
The inverse cosine function, written y = cos−1x or y = arccos x,
is the angle between 0 and π whose cosine is x.
In other words, y = cos−1 (x) if x = cos y and y is in [0,π]
The inverse cosine function
• The domain of y = cos x is restricted to
0 x 
• y = cos-1 x means that cos y = x.
• cos-1 x is the angle, between 0 and π, inclusive,
whose cosine value is x.
Find the exact value (in radians) of each of the
following:
1
cos   
2
1
 2
cos    
 2 
1
4.7 – Inverse Trig Functions
y = tan x
   
, 

2
2

The inverse tangent function, written y = tan−1x or y = arctan x,
is the angle between  π and π 2 whose tangent is x.
2
π
π

−1
In other words, y = tan x if x = tan y and
2< y < 2
The inverse tangent function
• The domain of y = tan x is restricted to


2
x

2
tan-1 x
(doesn’t include –π/2 and π/2, undefined at these)
• y=
means that tan y = x.
• tan-1x is the angle, between –π/2 and π/2,
whose tangent value is x.
Find the exact value (in radians) of
each of the following:
• tan (1)
• tan (
3)
Evaluating inverse functions
• For exact values, use your knowledge of the
unit circle.
• For approximate values, use your calculator
(be careful to watch your MODE).
Examples
Use your unit circle knowledge to
find an exact value in radians.
a ) sin 1
3
2
b ) cos 1 1
c ) tan 1  1 
More Examples
Use your calculator to find the value
in radians to four decimal places.
d ) cos
1
 0.46 
1 6
e) sin
5
Evaluating composite functions
• Composite functions come in two types:
1. The function is on the “inside”.
2. The inverse is on the “inside”.
• In either case, work from the “inside out”.
• Be sure to observe the restricted domains of the
functions you are dealing with.
• Sometimes the function and inverse will “cancel” each
other but, again, watch your restricted domains.
• For values not on the unit circle, draw a sketch and
use right triangle trigonometry.
Examples


a ) sin  sin 1 
4

7 

e) tan  cos 1

25 

b) tan  tan 1 6 

 5 
f ) sec sin 1    
 8 

3 

c ) tan 1  tan

4



 5 
g ) cos  tan 1    
 7 


 1 
d ) sin  cos 1   
 2 

Weird Examples
• Use a right triangle to write the following
expression as an algebraic expression:

1
cos sin 7 x

 1 2 
cos sin

x
