Download 1 Purpose of the week 2 Definitions and notations 3 Important points

Basic Calculus 1 (07.tex)
The 7th week
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Tuesday, February 24, 2015, 6:05 a.m.
Inverse trigonometric functions
Purpose of the week
We will define the inverse trigonometric functions, determining the generalized angle
whose trigonometric ratio is a given number.
Emphasis will be put on investigating
• domains and ranges, increasing and decreasing, and rates of variation,
• the particular values.
In order to do that, we always keep it in mind to draw the graphs of the functions.
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Definitions and notations
Let x be a number whose absolute value is smaller than or equal to 1. There are
infinitely many numbers y which satisfies the equality sin y = x. However, if we restrict
the range of y, say, to −π/2 ≤ y ≤ π/2, then the values of y with the equality is uniquely
determined. The function y as above is called the inverse sine function, and denoted by
y = arcsin x.
The domain and range are the set of the numbers whose absolute values are smaller than
or equal to 1 and π/2, respectively. Similarly, there are infinitely many numbers y which
satisfies the equality cos y = x. However, if we restrict the range of y, say, to 0 ≤ y ≤ π,
then the value of y with the equality is uniquely determined. The function y as above is
called the inverse cosine function, and denoted by
y = arccos x.
The domain and range are the set of the numbers whose absolute values are smaller
than or equal to 1, and the set of the numbers which are greater than or equal to 0 and
smaller than or equal to π, respectively.
Let x be an arbitrary number. There are infinitely many numbers y which satisfies
the equality tan y = x. However, if we restrict the range of y, say, to −π/2 ≤ y ≤ π/2,
then the values of y with the equality is uniquely determined. The function y as above
is called the inverse tangent function, and denoted by
y = arctan x.
The domain and range are the set of the numbers, and the set of numbers whose absolute
values are smaller than π/2, respectively.
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Important points for calculation
For the typical angles θ = 0, π/6, π/4, π/3, π/2, we are familiar with the values of the
trigonometric functions. Therefore, for such familiar numbers, the values of the inverse
trigonometric functions become the typical angles listed above.
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Basic Calculus 1 (07.tex)
4
Tuesday, February 24, 2015, 6:05 a.m.
Examples of the week
[1] Complete the following tables.
√
√
x
−1 − 3/2 −1/ 2 −1/2
arcsin x
arccos x
√
√
x
· · · − 3 −1 − 3
0
arctan x
0
1/2
√
1 3
1
√
1/ 2
√
3
√
3/2
1
···
[2] Prove the equality arcsin s + arccos s = π/2 for all numbers s with |s| ≤ 1.
Explanation Let θ = arcsin s. Then s = sin θ and −π/2 ≤ θ ≤ π/2. Since
sin θ = cos(π/2 − θ) and 0 ≤ π/2 − θ ≤ π, it follows that π/2 − θ = arccos s.
[3] Examine the increase, decrease, and convexity of the function y = arcsin x.
Explanation Let a < c < b, and put α = arcsin a, β = arcsin b, γ = arcsin c. Then
we have −π/2 ≤ α, β, γ ≤ π/2, and a = sin α, b = sin β, c = sin γ. Since the sine
function is increasing on the interval −π/2 ≤ x ≤ π/2, it follows that α < β. Define
ta = (γ − α)/(c − a), tb = (β − γ)/(b − a). Then t−1
= (sin γ − sin α)/(γ − α) and
a
=
(sin
β
−
sin
γ)/(β
−
γ).
Since
the
sine
function
is
concave (resp. convex) on the
t−1
b
−1
−1
−1
interval 0 ≤ x ≤ π/2 (resp. −π/2 ≤ x ≤ 0), it follows that t−1
a > tb (resp. ta < tb )
on the interval 0 ≤ x ≤ π/2 (resp. −π/2 ≤ x ≤ 0). Hence ta < tb (resp. ta > tb ) on the
interval 0 ≤ x ≤ π/2 (resp. −π/2 ≤ x ≤ 0).
We conclude that the function y = arcsin x is increasing, and, moreover, convex on
the interval 0 ≤ x ≤ π/2, and concave on the interval −π/2 ≤ x ≤ 0.
[4] State the shape of the graph of the function y = arcsin x as exactly as possible.
Explanation The analysis above gives us the following sketch of the graph.
y
y = arcsin x
pi/2
-1
0
1
-pi/2
2
x
Basic Calculus 1 (07.tex)
5
Tuesday, February 24, 2015, 6:05 a.m.
Exercises
[1] Prove the equality
arctan s + arctan
1
π
=
s
2
for all positive numbers s.
What can then be said if s is negative?
[2] Prove the following equalities applying the addition theorem for tangents.
(1) arctan
1
1
π
+ arctan = .
2
3
4
(2) arctan
1
1
1
+ arctan = arctan .
3
7
2
(3) arctan
1
1
1
π
+ arctan + arctan = .
2
5
8
4
[3] Examine the increase, decrease, and convexity of the following functions.
(1) y = arccos x.
(2) y = arctan x.
[4] State the shape of the graph of the following functions as exactly as possible.
(10) y = −3 arccos(x + 2).
(1) y = arccos x.
(2) y = arctan x.
(11) y =
(3) y = arcsin(−x).
1
arctan x.
2
(4) y = arccos(−x).
(12) y = arcsin(x − 1).
(5) y = arctan(−x).
(13) y = arccos(x + 2).
π
.
3
π
(7) y = arccos x − .
2
π
(8) y = arctan x + .
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(14) y = arctan(x − 2).
(6) y = arcsin x +
(15) y = 3 arcsin(x + 1) +
π
.
4
(16) y = −2 arccos(x − 2) −
(9) y = 2 arcsin(x − 1).
π
.
3
(17) y = 2 arctan(x + 2) + π.
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Basic Calculus 1 (07.tex)
6
Tuesday, February 24, 2015, 6:05 a.m.
Problems
Prove the equality
arcsin s + 2 arctan
r
π
1−s
=
1+s
2
for all numbers s with −1 < s ≤ 1.
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Summary of the week
Starting with the restriction of the domain of the trigonometric functions, we defined
the inverse trigonometric functions (the inverse sine function, the inverse cosine function,
the inverse tangent function).
The inverse sine function is an increasing function, and it is concave on the interval
−1 ≤ x ≤ 0, and convex on 0 ≤ x ≤ 1.
The inverse cosine function is a decreasing function, and it is convex on the interval
−1 ≤ x ≤ 0, and concave on 0 ≤ x ≤ 1.
The inverse tangent function is an increasing function, and it is convex on the interval
−1 ≤ x ≤ 0, and concave on 0 ≤ x ≤ 1.
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Supplementary talk
Similaly as the logarithm, we can understand the inverse trigonometric ratio as follows,
noting the relation with square roots.
Let M be an arbitrary number, and consider the numbers (the generalized angles)
whose values of sine or cosine are M . There exist infinitely many such numbers if
|M | ≤ 1 because of the periodicity of the functions, and none if |M | > 1. However,
when |M | ≤ 1, if we restrict the range of the numbers θ, say, to −π/2 ≤ θ ≤ π/2 or to
−π/2 ≤ θ ≤ π/2, then there is a unique value of θ satisfying the equality sin θ = M or
cos θ = M , respectively. Thus we denote these values by arcsin M, arccos M . As a tacit
understanding, we do not consider the inverse sine and the inverse cosine of the numbers
whose absolute values are greater than 1.
Let M be an arbitrary number, and consider the numbers (the generalized angles)
whose values of tangent are M . There exist infinitely many such numbers because of the
periodicity. However, if we restrict the range of the numbers θ, say, to −π/2 ≤ θ ≤ π/2,
then there is a unique value of θ satisfying the equality tan θ = M . Thus we denote
these values by arctan M .
Let A be the point (1, 0). For an arbitrary number s with −1 ≤ s ≤ 1, take a point
_
P(s, t) on the unit circle with t ≥ 0, and let ω be the directed length of the arc AP,
the length in consideration of the direction of the rotation from A to P. Then we have
ω = arccos s, s = cos ω. Similarly, for an arbitrary number t with −1 ≤ t ≤ 1, take a
point P(s, t) on the unit circle with s ≥ 0, and let ω be the directed length of the arc
_
AP. Then we have ω = arcsin t, t = sin ω.
Considering in this way, which of the trigonometric functions and the inverse trigonometric functions should be the more natural functions?
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