Download 4.7 Inverse Trigonometric Functions

Digital Lesson
Inverse Trigonometric
Functions
Inverse Sine Function
Recall that for a function to have an inverse, it must be a
one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test
and must be restricted to find its inverse.
y
y = sin x
1


2
x
1
Sin x has an inverse
function on this interval.
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The inverse sine function is defined by
y = arcsin x
if and only if
sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–/2 , /2].
Example:
a. arcsin 1  
2 6
b. sin 1 3  
2
3
 is the angle whose sine is 1 .
6
2
sin   3
3
2
This is another way to write arcsin x.
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Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse.
y
1

y = cos x

2
x
1
Cos x has an inverse
function on this interval.
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The inverse cosine function is defined by
y = arccos x if and only if
cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , ].
Example:

a.) arccos 1  
3
2 3
 5
1 
3
b.) cos  


 2  6
is the angle whose cosine is 1 .
2
cos 5   3
6
2
This is another way to write arccos x.
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Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse.
y
y = tan x

2
 3
2
3
2
x

2
Tan x has an inverse
function on this interval.
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The inverse tangent function is defined by
y = arctan x
if and only if
tan y = x.
Angle whose tangent is x
The domain of y = arctan x is (, ) .
The range of y = arctan x is [–/2 , /2].
Example:
a.) arctan 3  
3
6
 is the angle whose tangent is
b.) tan 1 3  
3
tan   3
3
6
3.
3
This is another way to write arctan x.
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Graphing Utility: Graph the following inverse functions.
Set calculator to radian mode.
a. y = arcsin x

–1.5
1.5
–
2
b. y = arccos x
–1.5
1.5
–

c. y = arctan x
–3
3
–
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Graphing Utility: Approximate the value of each expression.
Set calculator to radian mode.
a. cos–1 0.75
b. arcsin 0.19
c. arctan 1.32
d. arcsin 2.5
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Composition of Functions:
f(f –1(x)) = x
and (f –1(f(x)) = x.
Inverse Properties:
If –1  x  1 and – /2  y  /2, then
sin(arcsin x) = x and arcsin(sin y) = y.
If –1  x  1 and 0  y  , then
cos(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(arctan x) = x and arctan(tan y) = y.
Example: tan(arctan 4) = 4
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Example:
a. sin–1(sin (–/2)) = –/2
 
b. sin 1 sin 5 

3 
5 does not lie in the range of the arcsine function, –/2  y  /2.
3
y
However, it is coterminal with 5  2   
3
3
5
which does lie in the range of the arcsine
3
x

3
function.
 
 
sin 1 sin 5   sin 1 sin      


3 
3 
3
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
Example:

Find the exact value of tan arccos 2 .
3
adj 2
2
Let u = arccos , then cos u 
 .
3y
hyp 3
3
32  22  5
u


x
2
opp
2
tan arccos  tan u 
 5
3
adj
2
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