Transcript Section 4.6
Chapter 4
Trigonometric
Functions
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All rights reserved
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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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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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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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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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