Download 16 Inverse Trigonometric Functions

16
Inverse Trigonometric Functions
Concepts:
• Restricting the Domain of the Trigonometric Functions
• The Inverse Sine Function
• The Inverse Cosine Function
• The Inverse Tangent Function
• Using the Inverse Trigonometric Functions and the Periodicity Identities to Solve
Trigonometric Equations.
(Sections 7.4-7.5)
16.1
The Problems with Inverse Trigonometric Functions
Today we focus our attention on solving trigonometric equations. If we are going to have
any hope of doing this in a systematic manner that does not involve a lot of guess and check,
we need functions that can unravel or undo the trigonometric functions. In other words, we
need inverse trigonometric function. Why is the notion of an inverse trigonometric function
disturbing?
In order to define inverse trigonometric functions, we first need to
of the trigonometric functions so that the function is
.
There are multiple ways to
of a
trigonometric function so that we can define an inverse, but there are some restricted domains
that are better than others. A good restricted domain has the following characteristics:
1
Even with these restrictions, we can see that there are multiple ways to define a good
restricted domain. In the interest of consistency, uniformity, and transportability, most
mathematicians and scientists have agreed on fairly standard restricted domains for the
sine, cosine and tangent functions which will allow us to discuss the inverse sine (sin−1 or
arcsin) function, the inverse cosine (cos−1 or arccos) function, and the inverse tangent (tan−1
or arctan) function.
16.2
The Inverse Sine Function (sin−1 or arcsin)
The Restricted Domain of the Sine Function: [− π2 , π2 ]
y
1
- π2
•
•
π
2
x
-1
Definition 16.1 (The Inverse Sine Function)
For each v ∈ [−1, 1] there is a unique u ∈ − π2 , π2 such that sin(u) = v. We define
sin−1 (v) = u
when
sin(u) = v
h π πi
and u ∈ − ,
.
2 2
NOTE: arcsin can replace sin−1 in the previous definition.
The input of the sin function is
.
The output of the sin function is
.
The input of the sin−1 function is
.
2
The output of the sin−1 function is
.
Proposition 16.2
• sin−1 (sin(x)) = x if x ∈ − π2 , π2
• sin(sin−1 (x)) = x if x ∈ [−1, 1]
Example 16.3
For each of the following, find an exact value for the expression or state that it is undefined.
• sin−1
√ 3
2
• arcsin (1)
• arcsin − √12
√ • sin sin−1 23
• sin−1 sin
π
6
• sin−1 sin
5π
6
• sin(sin−1 (2))
• sin−1 (sin (1.3))
3
Example 16.4
Find the exact value of cos sin−1 − 35 .
16.3
The Inverse Cosine Function (cos−1 or arccos)
The Restricted Domain of the Cosine Function: [0, π]
y
1 •
π
π
2
-1
x
•
Definition 16.5 (The Inverse Cosine Function)
For each v ∈ [−1, 1] there is a unique u ∈ [0, π] such that cos(u) = v. We define
cos−1 (v) = u
when
cos(u) = v
and u ∈ [0, π] .
NOTE: arccos can replace cos−1 in the previous definition.
4
Proposition 16.6
• cos−1 (cos(x)) = x if x ∈ [0, π]
• cos(cos−1 (x)) = x if x ∈ [−1, 1]
Example 16.7
For each of the following, find an exact value for the expression or state that it is undefined.
√ • cos−1 − 23
• sin cos−1
1
2
• cos cos−1 − 12
• sin cos−1 − 12
• cos sin−1 − 12
• cos−1 cos
5π
6
Example 16.8 (Similar to Example 7 in Section 7.4 of your textbook.)
Write cos(sin−1 (x)) as an algebraic expression in x.
5
16.4
The Inverse Tangent Function (tan−1 or arctan)
The Restricted Domain of the Tangent Function: (− π2 , π2 )
y
5
4
3
2
1
0
0
− π2
-1
π
2
x
-2
-3
-4
-5
Definition 16.9 (The Inverse Tangent Function)
For each v ∈ (−∞, ∞) there is a unique u ∈ − π2 , π2 such that sin(u) = v. We define
tan−1 (v) = u
when
tan(u) = v
π π
and u ∈ − ,
.
2 2
NOTE: arctan can replace tan−1 in the previous definition.
Proposition 16.10
• tan−1 (tan(x)) = x if x ∈ − π2 , π2
• tan(tan−1 (x)) = x if x ∈ (−∞, ∞)
6
Example 16.11
For each of the following, find an exact value for the expression or state that it is undefined.
• tan−1
√ 3
• cos tan−1
8
)
5
• sin tan−1
8
)
5
Example 16.12
Write an algebraic expression for sec2 (tan−1 (x)) in terms of x.
7
16.5
Solving Some Trigonometric Equations
The inverse trigonometric functions can be used to help you find a solution for some trigonometric equations. If you wish to find all solutions, you will also need to recall the periodicity
identities.
Recall that
sin(π − x) = sin(x)
and
cos(−x) = cos(x).
Example 16.13
Find all solutions of the equations. Whenever possible, find exact solutions of the equation.
• sin(x) =
√
2
2
• 2 cos(3x) + 10 = 10 −
√
3
8
•
sin(5x) + 7
= 10
3
• tan(x) = − √13
• tan2 (x) + 1 = 5
9