Appendices - Appendix A MATHEMATICAL FORMULAS A.1...

Appendix A
MATHEMATICAL FORMULAS
A.1 TRIGONOMETRIC IDENTITIES
tan A =
sec A =
sin A
cos A'
1
cos A'
cot A =
1
esc A =
tan A
1
sin A
sin
2
A + cos
2
A = 1 , 1 + tan
2
A = sec
2
A
1 + cot
2
A = esc
2
A
sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B + sin A sin B
2 sin A sin B = cos (A - B) - cos (A + B)
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
sin A + sin B = 2 sin
B A -B
cos
. „ „ A + B A - B
sin A - sin B = 2 cos sin
A - B
A + B
cos A + cos B = 2 cos cos
A
n
^ . A + B A -B
cos A - cos B = - 2 sin sin
cos (A ± 90°) = +sinA
sin (A ± 90°) = ± cos A
tan (A ±90°) = -cot A
cos (A ± 180°) = -cos A
sin (A ± 180°) = -sin A
727

728 Appendix A
tan (A ± 180°) = tan A
sin 2A = 2 sin A cos A
cos 2A = cos
2
A - sin
2
A = 2 cos
2
A - 1 = 1 - 2 sin
2
A
tan A ± B
tan (A ± B) = ——
tan 2A =
1 + tan A tan B
2 tan A
1 - tan
2
A
sin A =
e
jA
- e~
iA
cos A =
2/ ' — " 2
e
jA
= cos A + y sin A (Euler's identity)
TT = 3.1416
1 rad = 57.296°
\.2 COMPUX VARIABLES
A complex number may be represented as
z = x + jy = r/l = re
je
= r (cos 0 + j sin
where x = Re z = r cos 0, y = Im z = r sin 0
7 =
l,
T
= -y,
The complex conjugate of z = z* = x — jy = r / - 0 = re
je
= r (cos 0 - j sin 0)
(e
j9
)" = e
jn6
= cos «0 + j sin «0 (de Moivre's theorem)
If Z\ = x, + jy
x
and z
2
= ^2 + i)
1
!. then z, = z
2
only if x
1
= JC
2
and j ! = y
2
.
Zi± Z
2
= (xi + x
2
) ± j(yi + y
2
)
or
nr
2
/o,

APPENDIX A
729
i j
y\
or
Z2
Vz = VxTjy = \Tre
m
= Vr /fl/2
2
n
= (x + /y)" = r" e
;nfl
= r
n
/nd (n = integer)
z
"» = (
X
+ yj,)"" =
r
1/n
e
^ " =
r
Vn
/din + 27rfc/n (t = 0, 1, 2,
, n -
In (re'*) = In r + In e
7
* = In r + jO + jlkir (k = integer)
A3 HYPERBOLIC FUNCTIONS
sinhx =
tanh x =
u ~ -
e
x
- e'
x
2
sinh x
cosh x
1
coshx =
COttlJt =
e
x
1
sechx =
tanhx
1
coshx
cosjx = coshx
coshyx = cosx
sinhx
sinyx — j sinhx,
sinhyx = j sinx,
sinh (x ± y) = sinh x cosh y ± cosh x sinh y
cosh (x ± y) = cosh x cosh y ± sinh x sinh y
sinh (x ± jy) = sinh x cos y ± j cosh x sin y
cosh (x ± jy) = cosh x cos y ±j sinh x sin y
sinh 2x sin 2y
tanh (x ± jy) = ± /
cosh 2x + cos 2y cosh 2x + cos 2y
cosh
2
x - sinh
2
x = 1
sech
2
x + tanh
2
x = 1
sin (x ± yy) = sin x cosh y ± j cos x sinh y
cos (x ± yy) = cos x cosh y + j sin x sinh y
L

730 • Appendix A
A.4 LOGARITHMIC IDENTITIES
If |
log xy = log x + log y
X
log - = log x - log y
log x" = n log x
log
10
x = log x (common logarithm)
log
e
x = In x (natural logarithm)
l,ln(l + x) = x
A.5 EXPONENTIAL IDENTITIES
e
x
=
where e == 2.7182
X ~f"
e
[e
In
x
2
2 !
4
V =
1" =
x
3
" 3!
+
e
x+y
X
x
4
4!
A.6 APPROXIMATIONS FOR SMALL QUANTITIES
If \x\ <Z 1,
(1 ± x)
n
== 1 ± ra
^ = 1 + x
In (1 + x) = x
sinx
sinx == x or hm = 1
>0 X
COS — 1
tanx — x

APPENDIX A «K 731
A.7 DERIVATIVES
If U = U(x), V = V(x), and a = constant,
dx dx
dx
dx dx
d\U
\
U
dx dx
V
2
~(aU
n
) = naU
n
~
i
dx
dx
U dx
d 1 dU
— In U =
dx U dx
d
v .t/, dU
— a = d In a —
dx dx
dx dx
dx dx
— sin U = cos U —
dx dx
d dU
—-cos U = -sin U —
dx dx
d , dU
— -tan U = sec £/ —
dx dx
d dU
— sinh U = cosh [/ —
dx dx
— cosh t/ = sinh {/ —
dx dx
d . dU
— tanh[/ = sech
2
t/ —
<ix dx
dx

732 Appendix A
A.8 INDEFINITE INTEGRALS
lfU= U(x), V = V(x), and a = constant,
a dx = ax + C
UdV=UV- | VdU (integration by parts)
U
n+l
U
n
dU = + C, n + - 1
n + 1
dU
U
= In U + C
a
u
dU = + C, a > 0, a
In a
e
u
dU = e
u
+C
e
ax
dx = - e
ax
+ C
a
xe
ax
dx = —r(ax - 1) + C
x e
ax
dx = — (
a
2
x
2
- lax + 2) + C
a'
In x dx = x In x — x + C
sin ax cfcc = — cos ax + C
a
cos ax ax = — sin ax + C
tan ax etc = - In sec ax + C = — In cos ax + C
a a
sec ax ax = — In (sec ax + tan ax) + C
a