Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson...
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Transcript of Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson...
![Page 1: Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfbe1a28abf838ca27a6/html5/thumbnails/1.jpg)
Section 7.4
Inverses of the Trigonometric
Functions
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
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Objectives
Find values of the inverse trigonometric functions. Simplify expressions such as sin (sin -1 x) and
sin -1 (sin x). Simplify expressions involving composition such as sin
(cos –1 1/2) without using a calculator. Simplify expressions such as sin arctan (a/b) by
making a drawing and reading off appropriate ratios.
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Inverse Sine Function
The graphs of an equation and its inverse are reflections of each other across the line y = x.
However, the inverse is not a function as it is drawn.
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Inverse Sine Function
We must restrict the domain of the inverse sine function.It is fairly standard to restrict it as shown here.
The domain is [–1, 1]
The range is [–π/2, π/2].
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Inverse Cosine Function
The graphs of an equation and its inverse are reflections of each other across the line y = x.
However, the inverse is not a function as it is drawn.
![Page 6: Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfbe1a28abf838ca27a6/html5/thumbnails/6.jpg)
Inverse Cosine Function
We must restrict the domain of the inverse cosine function.It is fairly standard to restrict it as shown here.
The domain is [–1, 1].
The range is [0, π].
![Page 7: Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfbe1a28abf838ca27a6/html5/thumbnails/7.jpg)
The graphs of an equation and its inverse are reflections of each other across the line y = x.
Inverse Tangent Function
However, the inverse is not a function as it is drawn.
![Page 8: Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfbe1a28abf838ca27a6/html5/thumbnails/8.jpg)
Inverse Tangent Function
We must restrict the domain of the inverse tangent function.It is fairly standard to restrict it as shown here.
The domain is (–∞, ∞).
The range is (–π/2, π/2).
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Inverse Trigonometric Functions
y sin 1 x
arcsin x, where x sin y
Function Domain Range
1, 1 2, 2
y cos 1 x
arccos x, where x cos y
1, 1 0,
y tan 1 x
arctan x, where x tan y
( , ) ( 2, 2)
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Graphs of the Inverse Trigonometric Functions
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Graphs of the Inverse Trigonometric Functions
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Example
Find each of the following function values.
a) sin 1 2
2b) cos 1
1
2
c) tan 1 3
2
Find such that sin = .2 2
In the restricted range [–π/2, π/2], the only number with sine of is π/4.2 2
Solution:
sin 1 2
2
4
, or 45º .
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Example (cont)
Find such that cos = –1/2.
In the restricted range [0, π], the only number with cosine of –1/2 is 2π/3.
cos 1 1
2
23
, or 120º .
b) cos 1 1
2
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Example (cont)
3 2.Find such that tan =
tan 1 3
3
6
, or 30º .
In the restricted range (–π/2, π/2), the only number with tangent of is –π/6. 3 2
c) tan 1 3
2
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Example
Approximate the following function value in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree.
Solution:
Press the following keys (radian mode):
cos 1 0.2689
Readout: Rounded: 1.8430
Rounded: 105.6º
Change to degree mode and press the same keys:
Readout:
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Composition of Trigonometric Functions
sin sin 1 x x, for all x in the domain of sin–1
cos cos 1 x x, for all x in the domain of cos–1
tan tan 1 x x, for all x in the domain of tan–1
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Example
Simplify each of the following.
Solution:
a) cos cos 1 3
2
b) sin sin 11.8
cos cos 1 3
2
3
2
a) Since is in the domain, [–1, 1], it follows that3 2
b) Since 1.8 is not in the domain, [–1, 1], we cannot evaluate the expression. There is no number with sine of 1.8. So, sin (sin–1 1.8) does not exist.
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Special Cases
sin 1 sin x x, for all x in the range of sin–1
cos 1 cos x x, for all x in the range of cos–1
tan 1 tan x x, for all x in the range of tan–1
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Example
Simplify each of the following.
Solution:
a) tan 1 tan6
b) sin 1 sin
34
tan 1 tan6
6
a) Since π/6 is in the range, (–π/2, π/2), it follows that
b) Since 3π/4 is not in the range, [–π/2, π/2], we cannot apply sin–1(sin x) = x.
sin 1 sin34
sin 1 2
2
4
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Example
Find
Solution:
sin cot 1 x
2
.
cot–1 is defined in (0, π), so consider quadrants I and II. Draw right triangles with legs x and 2, so cot = x/2.
sin cot 1 x
2
2
x2 4