Sullivan Algebra and Trigonometry: Section 7.1 The Inverse Sine, Cosine, and Tangent Functions Objectives of this Section • Find the Exact Value of the Inverse Sine, Cosine, and Tangent Functions • Find the Approximate Value of the Inverse Sine, Cosine, and Tangent Functions
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Sullivan Algebra and Trigonometry: Section 7.1
The Inverse Sine, Cosine, and Tangent Functions
Objectives of this Section
• Find the Exact Value of the Inverse Sine, Cosine, and Tangent Functions
• Find the Approximate Value of the Inverse Sine, Cosine, and Tangent Functions
Let f denote a one-to-one function y = f (x). The inverse of f, denoted f -1, is a function such that f -1(f (x)) = x for every x in the domain f and f (f -1(x)) = x for every x in the domain of f -1.
In other words, the function f maps each x in its domain to a unique y in its range. The inverse function f -1 maps each y in the range back to the x in the domain.
Recall the Definition of the Inverse Function
To find the inverse of the sine function, first examine the graph to see if the function is one - to - one, using the horizontal line test.
6.28 3.14 0 3.14 6.28
1.5
0.75
0.75
1.5 y = b
-1< b < 1
Since the sine function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.
x
y
2
2
-1
1
The inverse sine of x means
where and
y x x y
y x
sin sin1
2 21 1
sin sin 1
2u u u where
2
sin sin 1 1 1v v v where
Characteristics of y x sin 1
Domain of is the Range of
y x y x
x
sin sin :1
1 1
Range of is the Domain of
y x y x
y
sin sin :1
2 2
1 0 1
2,1
1,
2
2,1
1,2
y xsin
y x sin 1
2
3sin of eexact valu theFind 1y
22
2
3sin 1
22
2
3sin
3
y
.2
2sin of eexact valu theFind 1
22
2
2sin 1
22
2
2sin
4
y
To find the inverse of the cosine function, first examine the graph to see if the function is one - to - one, using the horizontal line test.
6.28 0 6.28
1.5
1.5
2 2
y = b -1 < y < 1
Since the cosine function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.
0 1.57 3.14
1.5
1.5
0 1,
, 1
The inverse cosine of x
means
where and
y x x y
y x
cos cos1
0 1 1
cos cos 1 0u u u where
cos cos 1 1 1v v v where
Domain of is the Range of
y x y x
x
cos cos :1
1 1
Characteristics of y x cos 1
Range of is the Domain of
y x y x
y
cos cos :1
0
1 0.38 1.76 3.14
1
0.38
1.76
3.14
, 1
1 0,
1,
0 1,
y xcos
y x cos 1
.2
3cos of eexact valu theFind 1
0 where
2
3cos 1
0 where2
3cos
6
.2
2cos of eexact valu theFind 1
0 where
2
2cos 1
0 where2
2cos
34
To find the inverse of the tangent function, first examine the graph to see if the function is one - to - one, using the horizontal line test.
7.85 0 7.85
5
5
Since the tangent function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.
1.57 0 1.57
5
5
2
2
The inverse tangent of x
means
where and
y x x y
y x
tan tan1
2 2
tan tan 1
2 2u u u where
vvv wheretantan 1
Characteristics of y x tan 1
Range of is the Domain of
2
y x y x
y
tan tan :1
2
x
xyxy
:tan of Range theis tan ofDomain 1
2 0 2
y xtan
y x tan 1
x 2
x 2
y 2
y 2
Find the exact value of tan . 1 3
tan 1 32 2
where
tan 32 2
where
3
Find the exact value of tan . 1 3
tan 1 32 2
where
tan 32 2
where
3
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