MATHEMATICAL SCIENCES
MATHEMATICS FOR ENGINEERING
20
FORMULA SHEET FS/1054-1055/19
2021-2021
1. Logarithm and exponential function
ax = ex ln a
The graph of y = ln x is
y
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y = ln x
0
x
1
2. Trigonometry and hyperbolic functions
Trigonometric identities
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ⌥ sin A sin B
tan A ± tan B
tan(A ± B) =
1 ⌥ tan A tan B
✓
◆
✓
◆
A+B
A B
sin A + sin B = 2 sin
cos
2
2
✓
◆
✓
◆
A+B
A B
sin A sin B = 2 cos
sin
2
2
✓
◆
✓
◆
A+B
A B
cos A + cos B = 2 cos
cos
2
2
✓
◆
✓
◆
A+B
A B
cos A cos B = 2 sin
sin
2
2
2 sin A cos B = sin(A
B) + sin(A + B)
2 cos A cos B = cos(A
B) + cos(A + B)
2 sin A sin B = cos(A
B)
cos(A + B)
The graphs of the three elementary trigonometric functions between
2⇡ and 2⇡ are:
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+1
2⇡
3⇡/2
⇡
⇡/2
⇡/2
1
Graph of cos ✓
1
⇡
✓
3⇡/2
2⇡
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+1
3⇡/2
2⇡
⇡
⇡/2
⇡
⇡/2
✓
3⇡/2
2⇡
1
Graph of sin ✓
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2
1
2⇡ 3⇡/2 ⇡
⇡/2
⇡
⇡/2
3⇡/2
✓
2⇡
1
2
Graph of tan ✓
Relationships for Plane Triangle
A
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c
B
b
a
C
a
b
c
=
=
sin A
sin B
sin C
a 2 = b2 + c 2
b2 = c2 + a2
2bc cos A,
c 2 = a 2 + b2
2ca cos B,
Hyperbolic functions
cosh x =
1 x
e +e
2
x
cosh2 x
,
sinh x =
1 x
e
2
e
x
.
sinh2 x = 1
The graphs of sinh x, cosh x and tanh x are
y
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1
y = cosh x
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y = sinh x
x
2
2ab cos C
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1
y = tanh x
x
Formulae involving the hyperbolic functions may be obtained from the corresponding trigonometric ones on
the previous page by replacing every product of sines and implied sines by minus the corresponding product
of hyperbolic sines. Thus, for instance, tan A tan B is an implied product of sines since it can be written
(sin A sin B)/(cos A cos B) and so, to obtain the corresponding formula, terms in tan A tan B would be replaced
by tanh A tanh B.
3. Di↵erentiation
function
derivative
tan x
sec2 x
cot x
cosec2 x
sec x
sec x tan x
cosec x
cosec x cot x
sinh x
cosh x
cosh x
sinh x
tanh x
sech2 x
1
tan
x
sin
1
x
cos
1
x
sinh
1
x
cosh
1
x
tanh
1
x
1
1 + x2
1
p
1 x2
1
p
1 x2
1
p
1 + x2
1
p
x2 1
1
1 x2
4. Complex numbers
Euler’s formula
e±j✓ = cos ✓ ± j sin ✓.
3
5. Vectors
Triple vector product
a ⇥ (b ⇥ c) = (a · c)b
6. Matrices
The inverse of a matrix A can be expressed
A
1
=
(a · b)c
adjA
|A|
7. Newton Raphson iteration formula
If xn is an approximation to a root of f (x) = 0 then a better approximation is usually
xn+1 = xn
f (xn )
f 0 (xn )
8. Approximate integration
In the following approximations fr = f (xr ) , where xr = x0 + rh .
1. Trapezium rule
Z
xn
x0
f (x) dx ⇡ h
✓
1
f0 + f1 + f2 + . . . + fn
2
1
1 + fn
2
◆
2. Simpson’s rule (in which n must be even)
Z
xn
x0
9. Power series
f (x) dx ⇡
1 ⇣
h f0 + 4 (f1 + f3 + . . . + fn
3
1)
+ 2 (f2 + f4 + . . . + fn
n(n 1) 2
(1 + x) = 1 + nx +
x + ... +
2!
n
where
✓ ◆
n(n
n
=
r
1)(n
2) . . . (n
r!
2)
✓ ◆
n
xr + . . . ,
r
r + 1)
.
If n is a positive integer the above series terminates and is convergent for all x.
If n is not a positive integer the series is infinite and converges for |x| < 1.
ex = 1 + x +
ln(1 + x) = x
cos x = 1
sin x = x
x2
xr
+ ... +
+ ...
2!
r!
x2
x3
+
2
3
x2
x4
+
2!
4!
x5
x3
+
3!
5!
. . . + ( 1)r+1
... +
... +
for all x
xr
+ ...
r
( 1)r x2r
+ ...
(2r)!
( 1)r x2r+1
+ ...
(2r + 1)!
4
for
1<x1
for all x
for all x
+ fn
⌘
Taylor’s theorem
f (x) = f (a) + (x
a)f 0 (a) +
where
Rn (x) =
a)2
(x
2!
f 00 (a) + . . . +
(x a)n+1 (n+1)
f
(c),
(n + 1)!
a)n (n)
f (a) + Rn (x) ,
n!
(x
a<c<x
10. Fourier series
The Fourier series expansion of a piecewise continuous periodic function f (t) of period T is
✓
◆
✓
◆◆
1 ✓
X
1
2n⇡t
2n⇡t
a0 +
an cos
+ bn sin
,
2
T
T
n=1
where
2
an =
T
T /2
Z
f (t) cos
T /2
✓
2n⇡t
T
◆
2
bn =
T
dt,
T /2
Z
✓
f (t) sin
T /2
2n⇡t
T
◆
dt,
11. Laplace transforms
Table
L {f (t)} ⌘ F (s)
f (t)
A
,
s
1
,
s a
A
eat
tn ,
n!
n = 1, 2 . . .
Re(s) > 0
!
,
s2 + ! 2
s
,
s2 + ! 2
cos !t
eat f (t)
Re(s) > 0
Re(s) > 0
F (s
tn f (t)
a)
( 1)n
df
dt
sF (s)
d2 f
dt2
f (t
Re(s) > a
,
sn+1
sin !t
H(t
Re(s) > 0
s2 F (s)
d F
dsn
f (0)
sf (0)
as
e
a)
a)H(t
n
s
a)
e
5
as
F (s)
df
(0)
dt
for
n = 0, 1, 2, . . .
12. Statistics
Mean
X=
variance
2
SX
=
1
n
8
n
<X
1 : i=1
n
1 X
Xi ,
n i=1
Xi 2
1
n
n
X
i=1
Xi
!2 9
=
;
.
Tables of the cumulative distribution function of the standard normal distribution are given below:
z
(z)
(z)
z
0.0
0.5
1.0
1.5
2.0
2.5
3.0
.5000
.6915
.8413
.9332
.9772
.9938
.9987
0.90
0.95
0.975
0.99
0.995
0.999
0.9995
1.2816
1.6449
1.9600
2.3263
2.5758
3.0902
3.2905
End of Formula Sheet
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