Handbook of mathematics for engineers and scienteists part 10 pdf

2.3. INVERSE TRIGONOMETRIC FUNCTIONS 31
Arccot x, are multi-valued. The following relations define the multi-valued inverse trigono-
metric functions:
sin

Arcsin x

= x,cos

Arccos x

= x,
tan

Arctan x

= x,cot

Arccot x

= x.
These functions admit the following verbal definitions: Arcsin x is the angle whose sine is
equal to x; Arccos x is the angle whose cosine is equal to x;Arctanx is the angle whose
tangent is equal to x; Arccot x is the angle whose cotangent is equal to x.
The principal (single-valued) branches ofthe inverse trigonometric functions are denoted
by
arcsin x ≡ sin
–1
x (arcsine is the inverse of sine),
arccos x ≡ cos
–1

x (arccosine is the inverse of cosine),
arctan x ≡ tan
–1
x (arctangent is the inverse of tangent),
arccot x ≡ cot
–1
x (arccotangent is the inverse of cotangent)
and are determined by the inequalities
–
π
2
≤ arcsin x ≤
π
2
, 0 ≤ arccos x ≤ π (–1 ≤ x ≤ 1);
–
π
2
<arctanx <
π
2
, 0 < arccot x < π (–∞ < x < ∞).
The following equivalent relations can be taken as definitions of single-valued inverse
trigonometric functions:
y =arcsinx,–1 ≤ x ≤ 1 ⇐⇒ x =siny,–
π
2
≤ y ≤
π
2

;
y = arccos x,–1 ≤ x ≤ 1 ⇐⇒ x =cosy, 0 ≤ y ≤ π;
y =arctanx,–∞ < x <+∞⇐⇒x =tany,–
π
2
< y <
π
2
;
y = arccot x,–∞ < x <+∞⇐⇒x =coty, 0 < y < π.
The multi-valued and the single-valued inverse trigonometric functions are related by
the formulas
Arcsin x =(–1)
n
arcsin x + πn,
Arccos x =
arccos x + 2πn,
Arctan x =arctanx + πn,
Arccot x = arccot x + πn,
where n = 0,
1, 2,
The graphs of inverse trigonometric functions are obtained from the graphs of the
corresponding trigonometric functions by mirror reflection with respect to the straight line
y = x (with the domain of each function being taken into account).
2.3.1-2. Arcsine: y =arcsinx.
This function is defined for all x [–1, 1] and its range is y [–
π
2
,
π

2
]. The arcsine is an
odd, nonperiodic, bounded function that crosses the axes Ox and Oy at the origin x = 0,
y = 0. This is an increasing function in its domain, and it takes its smallest value y =–
π
2
at
the point x =–1; it takes its largest value y =
π
2
at the point x = 1. The graph of the function
y =arcsinx is given in Fig. 2.10.
32 ELEMENTARY FUNCTIONS
2.3.1-3. Arccosine: y = arccos x.
This function is defined for all x [–1, 1], and its range is y [0, π]. It is neither odd nor
even. It is a nonperiodic, bounded function that crosses the axis Oy at the point y =
π
2
and
crosses the axis Ox at the point x = 1. This is a decreasing function in its domain, and at
the point x =–1 it takes its largest value y = π; at the point x = 1 it takes its smallest value
y = 0.Forallx in its domain, the following relation holds: arccos x =
π
2
–arcsinx.The
graph of the function y = arccos x is given in Fig. 2.11.
O
1
yxarcsin=
x

y
1
π
2
π
2
Figure 2.10. The graph of the function y =arcsinx.
O
1
π
x
y
yx= arccos
1
π
2
Figure 2.11. The graph of the function y = arccos x.
2.3.1-4. Arctangent: y =arctanx.
This function is defined for all x, and its range is y (–
π
2
,
π
2
). The arctangent is an odd,
nonperiodic, bounded function that crosses the coordinate axes at the origin x = 0, y = 0.
This is an increasing function on the real axis with no points of extremum. It has two
horizontal asymptotes: y =–
π
2

(as x → –∞)andy =
π
2
(as x → +∞). The graph of the
function y =arctanx is given in Fig. 2.12.
2.3.1-5. Arccotangent: y = arccot x.
This function is defined for all x, and its range is y (0, π). The arccotangent is neither odd
nor even. It is a nonperiodic, bounded function that crosses the axis Oy at the point y =
π
2
and does not cross the axis Ox. This is a decreasing function on the entire real axis with
no points of extremum. It has two horizontal asymptotes y = 0 (as x → +∞)andy = π (as
x →–∞). For all x, the following relation holds: arccot x =
π
2
–arctanx. The graph of the
function y = arccot x is given in Fig. 2.13.
O
1
x
y
1
yxarctan=
π
2
π
2
Figure 2.12. The graph of the function y =arctanx.
O
1

x
y
π
1
yxarccot=
π
2
Figure 2.13. The graph of the function y = arccot x.
2.3. INVERSE TRIGONOMETRIC FUNCTIONS 33
2.3.2. Properties of Inverse Trigonometric Functions
2.3.2-1. Simplest formulas.
sin(arcsin x)=x, cos(arccos x)=x,
tan(arctan x)=x, cot(arccot x)=x.
2.3.2-2. Some properties.
arcsin(–x) = – arcsin x,
arctan(–x) = – arctan x,
arccos(–x)=π – arccos x,
arccot(–x)=π – arccot x,
arcsin(sin x)=

x – 2nπ if 2nπ –
π
2
≤ x ≤ 2nπ +
π
2
,
–x + 2(n + 1)π if (2n + 1)π –
π
2

≤ x ≤ 2(n + 1)π +
π
2
,
arccos(cos x)=

x – 2nπ if 2nπ ≤ x ≤ (2n + 1)π,
–x + 2(n + 1)π if (2n + 1)π ≤ x ≤ 2(n + 1)π,
arctan(tan x)=x – nπ if nπ –
π
2
< x < nπ +
π
2
,
arccot(cot x)=x – nπ if nπ < x <(n + 1)π.
2.3.2-3. Relations between inverse trigonometric functions.
arcsin x+arccos x =
π
2
,arctanx+arccot x =
π
2
;
arcsin x =
⎧
⎪
⎪
⎪
⎪

⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
arccos
√
1 –x
2
if 0 ≤ x ≤ 1,
– arccos
√
1 –x
2
if –1 ≤ x ≤ 0,
arctan
x
√
1 –x
2
if –1 < x < 1,
arccot
√
1 –x
2

x
–π if –1 ≤ x < 0;
arccos x =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
arcsin
√
1 –x
2
if 0 ≤ x ≤ 1,
π –arcsin
√
1 –x
2
if –1 ≤ x ≤ 0,
arctan
√

1 –x
2
x
if 0 < x ≤ 1,
arccot
x
√
1 –x
2
if –1 < x < 1;
arctan x =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎩
arcsin
x
√
1 +x
2
for any x,
arccos
1
√
1 +x
2
if x ≥ 0,
– arccos
1
√
1 +x
2
if x ≤ 0,
arccot
1
x
if x > 0;
arccotx =
⎧
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
arcsin
1
√
1 +x
2
if x > 0,
π –arcsin
1
√
1 +x

2
if x < 0,
arctan
1
x
if x > 0,
π +arctan
1
x
if x < 0.
2.3.2-4. Addition and subtraction of inverse trigonometric functions.
arcsin x +arcsiny =arcsin

x

1 – y
2
+ y
√
1 – x
2

for x
2
+ y
2
≤ 1,
arccos x
arccos y = arccos


xy

(1 – x
2
)(1 – y
2
)

for x y ≥ 0,
34 ELEMENTARY FUNCTIONS
arctan x +arctany =arctan
x + y
1 – xy
for xy < 1,
arctan x –arctany =arctan
x – y
1 + xy
for xy >–1.
2.3.2-5. Differentiation formulas.
d
dx
arcsin x =
1
√
1 – x
2
,
d
dx
arccos x =–

1
√
1 – x
2
,
d
dx
arctan x =
1
1 + x
2
,
d
dx
arccot x =–
1
1 + x
2
.
2.3.2-6. Integration formulas.

arcsin xdx= x arcsin x +
√
1 – x
2
+ C,

arccos xdx= x arccosx –
√
1 – x

2
+ C,

arctan xdx= x arctanx –
1
2
ln(1 + x
2
)+C,

arccot xdx= x arccotx +
1
2
ln(1 + x
2
)+C,
where C is an arbitrary constant.
2.3.2-7. Expansion in power series.
arcsin x = x +
1
2
x
3
3
+
1×3
2×4
x
5
5

+
1×3×5
2×4×6
x
7
7
+ ···+
1×3×···× (2n – 1)
2×4×···× (2n)
x
2n+1
2n + 1
+ ··· (|x| < 1),
arctan x = x –
x
3
3
+
x
5
5
–
x
7
7
+ ···+(–1)
n–1
x
2n–1
2n – 1

+ ··· (|x| ≤ 1),
arctan x =
π
2
–
1
x
+
1
3x
3
–
1
5x
5
+ ···+(–1)
n
1
(2n – 1)x
2n–1
+ ··· (|x| > 1).
The expansions for arccos x and arccot x can be obtained from the relations arccos x =
π
2
–arcsinx and arccot x =
π
2
–arctanx.
2.4. Hyperbolic Functions
2.4.1. Definitions. Graphs of Hyperbolic Functions

2.4.1-1. Definitions of hyperbolic functions.
Hyperbolic functions are defi ned in terms of the exponential functions as follows:
sinh x =
e
x
– e
–x
2
,coshx =
e
x
+ e
–x
2
,tanhx =
e
x
– e
–x
e
x
+ e
–x
,cothx =
e
x
+ e
–x
e
x

– e
–x
.
The graphs of hyperbolic functions are given below.
2.4. HYPERBOLIC FUNCTIONS 35
2.4.1-2. Hyperbolic sine: y =sinhx.
This function is defined for all x and its range is the entire y-axis. The hyperbolic sine is an
odd, nonperiodic, unbounded function that crosses the axes Ox and Oy at the origin x = 0,
y = 0. This is an increasing function in its domain with no points of extremum. The graph
of the function y =sinhx is given in Fig. 2.14.
2.4.1-3. Hyperbolic cosine: y =coshx.
This function is defined for all x, and its range is y [1,+∞). The hyperbolic cosine is
a nonperiodic, unbounded function that crosses the axis Oy at the point 1 and does not
cross the axis Ox. This function is decreasing on the interval (–∞, 0) and is increasing on
the interval (0,+∞); it takes its smallest value y = 1 at x = 0. The graph of the function
y =coshx is given in Fig. 2.15.
O
1
1
2
x
y
1
1
2
yxsinh=
Figure 2.14. The graph of the function y =sinhx.
O
1
1

2
2
3
4
x
y
1
2
yxcosh=
Figure 2.15. The graph of the function y =coshx.
2.4.1-4. Hyperbolic tangent: y =tanhx.
This function is defi ned for all x, and its range is y (–1, 1). The hyperbolic tangent is an
odd, nonperiodic, bounded function that crosses the coordinate axes at the origin x = 0, y = 0.
This is an increasing function on the entire real axis and has two horizontal asymptotes:
y =–1 (as x → –∞)andy = 1 (as x → +∞). The graph of the function y =tanhx is given
in Fig. 2.16.
2.4.1-5. Hyperbolic cotangent: y =cothx.
This function is defined for all x ≠0, and its range consists of all y (–∞,–1)andy (1,+∞).
The hyperbolic cotangent is an odd, nonperiodic, unbounded function that does not cross
the coordinate axes. This is a decreasing function on each of the semiaxes of its domain;
it has no points of extremum and does not cross the coordinate axes. It has two horizontal
asymptotes: y =–1 (as x → –∞)andy = 1 (as x → +∞). The graph of the function
y =cothx is given in Fig. 2.17.
36 ELEMENTARY FUNCTIONS
O
1
1
2
x
y

1
1
2
yxtanh=
Figure 2.16. The graph of the function y =tanhx.
O
1
x
y
1
yxcoth=
Figure 2.17. The graph of the function y =cothx.
2.4.2. Properties of Hyperbolic Functions
2.4.2-1. Simplest relations.
cosh
2
x –sinh
2
x = 1,
sinh(–x)=–sinhx,
tanh x =
sinh x
cosh x
,
tanh(–x)=–tanhx,
1 –tanh
2
x =
1
cosh

2
x
,
tanh x coth x = 1,
cosh(–x)=coshx,
coth x =
cosh x
sinh x
,
coth(–x)=–cothx,
coth
2
x – 1 =
1
sinh
2
x
.
2.4.2-2. Relations between hyperbolic functions of single argument (x ≥ 0).
sinh x =

cosh
2
x – 1 =
tanh x
√
1 –tanh
2
x
=

1
√
coth
2
x – 1
,
cosh x =

sinh
2
x + 1 =
1
√
1 –tanh
2
x
=
coth x
√
coth
2
x – 1
,
tanh x =
sinh x
√
sinh
2
x + 1
=

√
cosh
2
x – 1
cosh x
=
1
coth x
,
coth x =
√
sinh
2
x + 1
sinh x
=
cosh x
√
cosh
2
x – 1
=
1
tanh x
.
2.4.2-3. Addition formulas.
sinh(x y)=sinhx cosh y sinh y cosh x,cosh(x y)=coshx cosh y sinh x sinh y,
tanh(x
y)=
tanh x tanh y

1 tanh x tanh y
,coth(x y)=
coth x coth y 1
coth y coth x
.
2.4. HYPERBOLIC FUNCTIONS 37
2.4.2-4. Addition and subtraction of hyperbolic functions.
sinh x sinh y = 2 sinh

x
y
2

cosh

x
y
2

,
cosh x +coshy = 2 cosh

x + y
2

cosh

x – y
2


,
cosh x –coshy = 2 sinh

x + y
2

sinh

x – y
2

,
sinh
2
x –sinh
2
y =cosh
2
x –cosh
2
y =sinh(x + y)sinh(x – y),
sinh
2
x +cosh
2
y =cosh(x + y)cosh(x – y),
(cosh x
sinh x)
n
=cosh(nx) sinh(nx),

tanh x
tanh y =
sinh(x y)
cosh x cosh y
,cothx coth y =
sinh(x y)
sinh x sinh y
,
where n = 0,
1, 2,
2.4.2-5. Products of hyperbolic functions.
sinh x sinh y =
1
2
[cosh(x + y)–cosh(x – y)],
cosh x cosh y =
1
2
[cosh(x + y)+cosh(x – y)],
sinh x cosh y =
1
2
[sinh(x + y)+sinh(x – y)].
2.4.2-6. Powers of hyperbolic functions.
cosh
2
x =
1
2
cosh 2x+

1
2
,
cosh
3
x =
1
4
cosh 3x+
3
4
cosh x,
cosh
4
x =
1
8
cosh 4x+
1
2
cosh 2x+
3
8
,
cosh
5
x =
1
16
cosh 5x+

5
16
cosh 3x+
5
8
cosh x,
sinh
2
x =
1
2
cosh 2x–
1
2
,
sinh
3
x =
1
4
sinh 3x–
3
4
sinh x,
sinh
4
x =
1
8
cosh 4x–

1
2
cosh 2x+
3
8
,
sinh
5
x =
1
16
sinh 5x–
5
16
sinh 3x+
5
8
sinh x,
cosh
2n
x =
1
2
2n–1
n–1

k=0
C
k
2n

cosh[2(n –k)x]+
1
2
2n
C
n
2n
,
cosh
2n+1
x =
1
2
2n
n

k=0
C
k
2n+1
cosh[(2n –2k +1)x],
sinh
2n
x =
1
2
2n–1
n–1

k=0

(–1)
k
C
k
2n
cosh[2(n –k)x]+
(–1)
n
2
2n
C
n
2n
,
sinh
2n+1
x =
1
2
2n
n

k=0
(–1)
k
C
k
2n+1
sinh[(2n –2k +1)x].
Here, n = 1, 2, ;and C

k
m
are binomial coefficients.