Transcript Section 4.6

Chapter 4
Trigonometric
Functions
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All rights reserved
© 2010 Pearson Education, Inc. All rights reserved
1
SECTION 4.6
Inverse Trigonometric Functions
OBJECTIVES
1
2
3
4
5
Graph and apply the inverse sine function.
Graph and apply the inverse cosine function.
Graph and apply the inverse tangent function.
Evaluate inverse trigonometric functions
using a calculator.
Find exact values of composite functions
involving the inverse trigonometric functions.
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2
INVERSE SINE FUNCTION
If we restrict the domain of y = sin x to the
  
interval   ,  ,
 2 2
then it is a oneto-one function
and its inverse
is also a
function.
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3
INVERSE SINE FUNCTION




The inverse function for y = sin x,   x 
2
2
is called the inverse sine, or
arcsine,
The graph is
function.
obtained by
reflecting the
graph of
y = sin x, for

x
2
2
in the line
y = x.
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4
INVERSE SINE FUNCTION
y = sin–1 x means sin y = x,
where –1 ≤ x ≤ 1 and 

2
y

2
.
Read y = sin–1 x as “y equals inverse sine at x.”
The domain of y = sin–1 x is [–1, 1].
The range of y =
sin–1
  
x is   ,  .
 2 2
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EXAMPLE 1
Finding the Exact Value for y = sin–1 x
Find the exact values of y.
1
3
1
1 
1
b. y  sin    c. y  sin 3
a. y  sin
 2
2
Solution
3
3


a. y  sin
means sin y 
,  y .
2
2
2
2

3

Since sin 
and
is in the interval
3
2
3
3 
  
1
we
have
y

sin

.

,
,
 2 2 
2
3
1
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EXAMPLE 1
Finding the Exact Value for y = sin–1 x
Solution continued
1


 1
b. y  sin    means sin y   ,   y  .
 2
2
2
2
1

 
Since sin      and  is in the interval
 6
2
6
1

  
1 
we
have
y

sin



.

,
,


 2 2 
 2
6
1
c. Since 3 is not in the domain of the inverse
sine function, which is [–1, 1], sin–1 3 does
not exist.
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INVERSE COSINE FUNCTION
If we restrict the domain of y = cos x to the
interval [0, π],
then it is a oneto-one function
and its inverse
is also a
function.
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INVERSE COSINE FUNCTION
The inverse function for y = cos x, 0  x  
is called the
inverse cosine, or
arccosine,
function.
The graph is
obtained by
reflecting the graph
of y = cos x, with
0  x   in the
line y = x.
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INVERSE COSINE FUNCTION
y = cos–1 x means cos y = x,
where –1 ≤ x ≤ 1 and 0  y   .
Read y = cos–1 x as “y equals inverse cosine at x.”
The domain of y = cos–1 x is [–1, 1].
The range of y = cos–1 x is  0,  .
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EXAMPLE 2
Finding the Exact Value for cos–1 x
Find the exact values of y.
1
2
1 
1
b. y  cos   
a. y  cos
 2
2
Solution
2
2
a. y  cos
means cos y 
, 0  y  .
2
2

2

Since cos 
and 0    ,
4
2
4
2 
1
we have y  cos
 .
2
4
1
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EXAMPLE 2
Finding the Exact Value for cos–1 x
Solution continued
1
 1
b. y  cos    means cos y   , 0  y   .
 2
2
1
2
1
2
Since cos
  and 0 
 ,
3
2
3
 1  2
we have y  cos    
.
 2
3
1
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INVERSE TANGENT FUNCTION
If we restrict the domain of y = tan x to the interval
   , then
  , 
2 2
it is a one-toone function
and its inverse
is also a
function.
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13
INVERSE TANGENT FUNCTION
The inverse function for y = tan x, 

x

2
2
is called the inverse
tangent, or
arctangent, function.
The graph is obtained
by reflecting the graph
of y = tan x, with


x

2
2
line y = x.
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, in the
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INVERSE TANGENT FUNCTION
y = tan–1 x means tan y = x,


where –∞ ≤ x ≤ ∞ and   y  .
2
2
Read y = tan–1 x as “y equals inverse tangent at x.”
The domain of y = tan–1 x is [–∞, ∞].
The range of y =
tan–1
  
x is   ,  .
 2 2
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EXAMPLE 3
Finding the Exact Value for tan–1 x
Find the exact values of y.
1
a. y  tan 0
b. y  tan
1
 3
Solution
a. y  tan 1 0
Since tan 0  0 and 

2
0

2
,
1
we have y  tan 0  0.
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EXAMPLE 3
Finding the Exact Value for tan–1 x
Solution continued
b. y  tan
1
 3

 
 
Since tan      3 and     ,
 3
2
3 2
we have y  tan
1

 3   3 .
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INVERSE COTANGENT FUNCTION
y = cot–1 x means cot y = x,
where –∞ ≤ x ≤ ∞ and 0  y   .
INVERSE COSECANT FUNCTION
y = csc–1 x means csc y = x,
where |x| ≥ 1 and 

2
y

2
, y  0.
INVERSE SECANT FUNCTION
y = sec–1 x means sec y = x,

where |x| ≥ 1 and 0  y   , y  .
2
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EXAMPLE 4
Finding the Exact Value for csc–1 x
Find the exact for y = csc−1 2.
Solution
Since
and
we have
y = csc−1 2 =
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USING A CALCULATOR WITH INVERSE
TRIGONOMETRIC FUNCTIONS
To find
csc–1
1
x find sin
.
x
To find
sec–1
1
x find cos
.
x
1
1
1
.
To find
x start by finding tan
x
If x ≥ 0, this is the correct value.
If x < 0, add π to get the correct value.
cot–1
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1
20
EXAMPLE 5
Using a Calculator to Find the Values of
Inverse Functions
Use a calculator to find the value of y in radians
rounded to four decimal places.
a. y  sin 1 0.75
b. y  cot 1 2.8
c. y = cot−1 (−2.3)
Solution
Set the calculator to Radian mode.
a. y  sin 1 0.75  0.8481
1
1  1 
b. y  cot 2.8  tan 
  0.3430
 2.8 
c. y = cot−1 (−2.3) = π + tan−1
≈ 2.7315
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EXAMPLE 6
Using a Calculator to Find the Values of
Inverse Functions
Use a calculator to find the value of y in degrees
rounded to four decimal places.
a. y  tan 1 0.99
b. y  sec 1 25
c. y = cot−1 (−1.3)
Solution
Set the calculator to Degree mode.
1
a. y  sin 0.99  44.7121
1
1  1 
b. y  sec 25  cos    87.7076
 25 
c. y = cot−1 (−1.3) = 180º + tan−1
≈ 142.4314º
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COMPOSITION OF TRIGONOMETRIC AND
INVERSE TRIGONOMETRIC FUNCTIONS
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EXAMPLE 7
Finding the Exact Value of sin−1 (sin x) and
cos−1 (cos x)
Find the exact value of
Solution
a. Because
we have
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EXAMPLE 7
Finding the Exact Value of sin−1 (sin x) and
cos−1 (cos x)
Solution continued
is not in the interval [0, π], but
b.
cos
.
So,
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EXAMPLE 8
Finding the Exact Value of a Composite
Trigonometric Expression
Find the exact value of
 1  1  
 1 2 
b. sin  cos    
a. cos  tan

 4

3

Solution
a. Let  represent the radian measure of the
2
 

angle in   ,  , with tan   .
 2 2
3
Since tan  is positive,  must be positive,

1 2
  tan
and 0    .
3
2
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EXAMPLE 8
Finding the Exact Value of a Composite
Trigonometric Expression
Solution continued
2
  tan
3
1
y 2
tan    .
x 3
So x = 3 and y = 2.
r  x 2  y2
 3 2
2
2
 9  4  13
3
3
3 13
 1 2 
cos  tan
 cos  



3
r
13
13
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EXAMPLE 8
Finding the Exact Value of a Composite
Trigonometric Expression
Find the exact value of
 1  1  
b. sin  cos    
 4

Solution
b. Let  represent the radian measure of the
1
angle in 0,  , with cos   .
4
Since cos  is negative, we have
1

1 
  cos    and     .
 4
2
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EXAMPLE 8
Finding the Exact Value of a Composite
Trigonometric Expression
Solution continued
x 1
1


1
  cos    cos  r  4
 4
So x = –1 and r = 4.
2
2
2
r x y
4  1  y
2
2
2
15  y 2
y  15
y
15
 1  1  
sin  cos      sin   
 4
r
4

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EXAMPLE 9
Finding the Rotation Angle for a Security
Camera
A security camera is to be installed 20 feet away
from the center of a jewelry counter. The
counter is 30 feet long.
What angle, to the
nearest degree, should
the camera rotate
through so that it scans
the entire counter?
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EXAMPLE 9
Finding the Rotation Angle for a Security
Camera
Solution
The counter center , the camera , and a counter
end form a right triangle.
The angle at vertex A
is
where θ is the
angle through which
the camera rotates.
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EXAMPLE 9
Finding the Rotation Angle for a Security
Camera
Solution continued
Set the camera to 74º rotate through to scan the
entire counter.
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