University of Edinburgh

Discipline of Civil and
Environmental Engineering

Mathematics Databook

April 2011

Contents1
1.

TRIGONOMETRIC FUNCTIONS............................................................................................................. 1

2.

HYPERBOLIC FUNCTIONS...................................................................................................................... 1

3.

GEOMETRICAL FORMULAE.................................................................................................................. 2

4.

LIMITS .......................................................................................................................................................... 3

5.

SERIES........................................................................................................................................................... 3

6.

DIFFERENTIATION ................................................................................................................................... 6

7.

PARTIAL DIFFERENTIATION................................................................................................................. 6

8.

INTEGRATION ............................................................................................................................................ 7

9.

NUMERICAL ANALYSIS........................................................................................................................... 8

10.

MATRIX ALGEBRA.................................................................................................................................. 12

11.

VECTOR PRODUCTS............................................................................................................................... 14

12.

COMPLEX VARIABLES .......................................................................................................................... 15

13.

LAPLACE TRANSFORMS ....................................................................................................................... 15

14.

FOURIER SERIES ..................................................................................................................................... 17

15.

STATISTICS ............................................................................................................................................... 18

16.

ORDINARY DIFFERENTIAL EQUATIONS ......................................................................................... 26

1.

Trigonometric Functions

sin( A ± B ) = sin A cos B ± cos A sin B tan( A ± B ) =

cos( A ± B ) = cos A cos B m sin A sin B

tan A ± tan B
1 m tan A tan B

sin A + sin B = 2 sin

A+ B
A− B cos 2
2

cos A + cos B = 2 cos

A+ B
A− B cos 2
2

sin A sin B = 1 cos( A − B ) − cos( A + B )
2

sin A − sin B = 2 cos

A+ B
A− B sin 2
2

cos A − cos B = −2 sin

A+ B
A− B sin 2
2

cos A cos B = 1 cos( A + B ) + cos( A − B )
2

11
( AA−+ BB)) ]
[sin(AA+−BB))−+sin
sin A cos BB == 2 sin( sin( 2

sin 2 x = 1 1 − cos 2 x

cos 2 x = 1 1 + cos 2 x

sin 3 x = 1 3 sin x − sin 3 x

cos 3 x = 1 3 cos x + cos 3 x

2

2

4

4

sin 2 x + cos2 x = 1 sin x =

e ix − e − ix
2i

2.

Hyperbolic Functions

cosh x =

e x + e− x
2

cos x =

sinh x =

e ix + e − ix
2

e x − e− x
2

cosh ix = cos x

cos ix = cosh x

sinh ix = i sin x

sin ix = i sinh x

cosh( x ± y ) = cosh x cosh y ± sinh x sinh y

sinh( x ± y ) = sinh x cosh y ± cosh x sinh y

cosh( x ± iy ) = cosh x cos y ± i sinh x sin y

sinh( x ± iy ) = sinh x cos y ± i cosh x sin y

cosh 2 x − sinh 2 x = 1

1

3.

Geometrical formulae

3.1

Triangles
B

c

a

A

C

b

∆ = 1 bc sin A = 1 ca sin B = 1 ab sin C

Area of triangle:

2

2

2

s ( s − a )( s − b )( s − c ) where 2s = a + b + c

or

sin A sin B sin C
=
= a b c Sine Rule:

a 2 = b 2 + c 2 − 2bc cos A

Cosine Rule:

b 2 = c 2 + a 2 − 2 ca cos B c 2 = a 2 + b 2 − 2 ab cos C

3.2

Circles

Circle radius r:

Perimeter = 2πr

Length of arc