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Section 7.4
Inverses of the Trigonometric
Functions
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
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.
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.
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].
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.
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].
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.
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).
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)
Graphs of the Inverse Trigonometric Functions
Graphs of the Inverse Trigonometric Functions
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 .
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
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
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:
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
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.
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
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
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