Download Lecture 5: Inverse Trigonometric Functions

Lecture 5: Inverse Trigonometric Functions
5.1 The inverse sine function
The function f (x) = sin(x) is not one-to-one on (−∞, ∞), but is on − π2 , π2 . Moreover,
f still has range
[−1, 1] when restricted to this interval. Hence it is reasonable to restrict
π π
f to − 2 , 2 to obtain an inverse for the sine function.
Definition We call the inverse of the sine function, when restricted to − π2 , π2 , the
arcsine function, with value at a point x denoted by arcsin(x) or sin−1 (x).
In other words,
π
π
≤y≤ .
2
2
Note that the arcsine function has domain [−1, 1] and range − π2 , π2 .
y = sin−1 (x) if and only if sin(y) = x and −
Example
We have
sin−1 (0) = 0,
π
,
2
π
sin−1 (−1) = − ,
2
sin−1 (1) =
and
−1
sin
1
−
2
π
=− .
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Note that sin(sin−1 (x)) = x for any −1 ≤ x ≤ 1 and sin−1 (sin(x)) = x for any − π2 ≤ x ≤
π
2 . However, the latter equality does not hold for other values of x. For example,
−1
sin
sin
7π
6
= sin
−1
1
−
2
π
=− .
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Since for x in [−1, 1] we have sin(sin−1 (x)) = x, it follows that
d
d
sin(sin−1 (x)) =
x.
dx
dx
Hence
cos(sin−1 (x))
d
sin−1 (x) = 1,
dx
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Lecture 5: Inverse Trigonometric Functions
and so
5-2
d
1
sin−1 (x) =
.
dx
cos(sin−1 (x))
Now if y = sin−1 (x), then − π2 ≤ y ≤
π
2
and sin(y) = x. Thus
cos2 (y) = 1 − sin2 (y) = 1 − x2 .
Since cos(y) ≥ 0 for − π2 ≤ y ≤
π
2,
we have
cos(sin−1 (x)) = cos(y) =
p
1 − x2 .
Thus we have the following proposition.
Proposition
Example
d
1
sin−1 (x) = √
for all −1 < x < 1.
dx
1 − x2
We have
d
6x
sin−1 (3x2 ) = √
.
dx
1 − 9x4
Note that f (x) = sin−1 (x) is an increasing function on [−1, 1], with graph as shown below.
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Graphs of y = sin(x) and y = sin−1 (x)
We now have the integration formula
Z
1
√
dx = sin−1 (x) + c.
2
1−x
1
1.5
Lecture 5: Inverse Trigonometric Functions
Example
5-3
Using the substitution u = 2x, we have
Z
0
1
4
Z
1
2
1
du
1 − u2
0
12
1
−1
= sin (u)
2
0
1
1
1
= sin−1
− sin−1 (0)
2
2
2
π
=
.
12
1
1
√
dx =
2
1 − 4x2
√
5.2 The inverse tangent function
Note that the function f (x) = tan(x) is one-to-one if its domain is restricted to − π2 , π2 .
Moreover, f still has range
(−∞, ∞) when restricted to this interval. Hence it is reasonable
π π
to restrict f to − 2 , 2 to obtain an inverse for the tangent function.
Definition We call the inverse of the tangent function, when restricted to − π2 , π2 , the
arctangent function, with value at a point x denoted by arctan(x) or tan−1 (x).
In other words,
y = tan−1 (x) if and only if tan(y) = x and −
π
π
<y< .
2
2
Note that the arctangent function has domain (−∞, ∞) and range − π2 , π2 .
Example
We have
tan−1 (0) = 0,
tan−1 (1) =
and
π
,
4
√
π
tan−1 (− 3) = − .
3
Since tan(tan−1 (x)) = x for all x in (−∞, ∞), we have
d
d
tan(tan−1 (x)) =
x.
dx
dx
Hence
sec2 (tan−1 (x))
d
tan−1 (x) = 1,
dx
Lecture 5: Inverse Trigonometric Functions
and so
5-4
d
1
tan−1 (x) =
.
2
dx
sec (tan−1 (x))
Now if y = tan−1 (x), then − π2 < y <
π
2
and tan(y) = x. Thus
sec2 (y) = 1 + tan2 (y) = 1 + x2 .
Thus we have the following proposition.
Proposition
Example
d
1
tan−1 (x) =
.
dx
1 + x2
If f (x) = 4 tan−1
x
2
, then
0
f (x) =
1+
4
x 2
1
8
=
.
2
4 + x2
2
Note that f (x) = tan−1 (x) is an increasing function on (−∞, ∞) with
lim tan−1 (x) =
x→∞
and
π
2
π
lim tan−1 (x) = − .
x→−∞
2
Hence the lines y =
π
2
and y = − π2 are horizontal asymptotes for the graph of f . Moreover,
f 00 (x) = −(1 + x2 )−2 (2x) = −
2x
,
(1 + x2 )2
from which it follows that the graph of f is concave upward on (−∞, 0) and concave
downward on (0, ∞). The graph is shown below.
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2
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Graphs of y = tan(x) and y = tan−1 (x)
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Lecture 5: Inverse Trigonometric Functions
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We now have the integration formula
Z
1
dx = tan−1 (x) + c.
2
1+x
Example
Z
Using the substitution u = x2 , we have
1
1
dx =
2
4+x
4
Z
1
1
dx =
2
2
1 + x2
Z
1
1
1
−1
−1 x
du
=
tan
(u)
+
c
=
tan
+ c.
1 + u2
2
2
2
5.3 The inverse secant function
Note that the function f (x) = sec(x) is one-to-one if its domain is restricted to
h π 3π 0,
∪ π,
.
2
2
Moreover, f still has range (−∞, −1] ∪ [1, ∞) when restricted to this domain. Hence it is
reasonable to restrict f to this domain to obtain an inverse for the secant function.
Definition
We call the inverse of the secant function, when restricted to
h π 3π 0,
∪ π,
,
2
2
the arcsecant function, with value at a point x denoted by arcsec(x) or sec−1 (x).
In other words,
y = sec−1 (x) if and only if sec(y) = x and either 0 ≤ y <
Note that the arcsecant function has domain
(−∞, −1] ∪ [1, ∞)
and range
h
Example
π
3π
0,
∪ π,
.
2
2
We have
sec−1 (1) = 0,
sec−1 (−1) = π,
3π
π
or π ≤ y <
.
2
2
Lecture 5: Inverse Trigonometric Functions
and
sec−1 (−2) =
5-6
4π
.
3
Since sec(sec−1 (x)) = x, we have
d
d
sec(sec−1 (x)) =
x.
dx
dx
Hence
sec(sec−1 (x)) tan(sec−1 (x))
and so
d
sec−1 (x) = 1,
dx
d
1
sec−1 (x) =
.
dx
x tan(sec−1 (x))
Now if y = sec−1 (x), then
tan2 (y) = sec2 (y) − 1 = x2 − 1.
Since
tan(y) ≥ 0 for y in the range of the arcsecant function, we have tan(sec−1 (x)) =
√
x2 − 1. Thus we have the following proposition.
Proposition
For all x ≥ 1 or x ≤ −1,
d
1
sec−1 (x) = √
.
dx
x x2 − 1
The previous proposition gives us the integration formula
Z
1
√
dx = sec−1 (x) + c.
2
x 1−x
The graph of f (x) = sec−1 (x) is shown below. Notice how the vertical asymptotes of
g(x) = sec(x) are transformed into horizontal asymptotes for f .
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Graphs of y = sec(x) and y = sec−1 (x)
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Lecture 5: Inverse Trigonometric Functions
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5.4 The remaining inverse trigonometric functions
The remaining inverse trigonometric functions are not used as frequently as the three we
have considered above. They are defined as follows:
y = cos−1 (x) if and only if cos(y) = x and 0 ≤ y ≤ π,
y = cot−1 (x) if and only if cot(y) = x and 0 < y < π,
and
y = csc−1 (x) if and only if csc(y) = x and 0 < y ≤
π
3π
or π < y ≤
.
2
2
One may show that
d
1
cos−1 (x) = − √
,
dx
1 − x2
d
1
cot−1 (x) = −
,
dx
1 + x2
and
d
1
csc−1 (x) = − √
.
dx
x x2 − 1
Note that these are just the negations of the derivatives of the inverses of the corresponding
co-functions. In particular, they do not give us any new integration formulas. Moreover,
this implies that these functions differ by only a constant from the inverses of their corresponding co-functions.