Mathematical Formulae Book (24pgs).pdf

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MATHEMATICAL FORMULAE

Uni QA

41

T594 2016

NG, S. N. SZE & K. L. CHIEW

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MATHEMATICAL FORMULAE

BOOK

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MATHEMATICAL

FoivIuLaE

BOOK

W. K. TIONG, S. N. SZE & K. L. CHIEW

Universiti Malaysia Sarawak Kota Samarahan

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0 W. K. Tiong, S. N. Sze & K. L Chiew, 2016

reproduced, stored in retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,

recording or otherwise, without the prior permission of the publisher.

Published in Malaysia by UNIMAS Publisher,

Universiti Malaysia Sarawak.

94300 Kota Samarahan, Sarawak, Malaysia.

Printed in Malaysia by

Percetakan Nasional Malaysia Berhad, Jalan Tun Abang Haji Openg,

93554 Kuching,

Sarawak, Malaysia.

Perpustakaan Negara Malaysia Cataloguing-in-Publication Data Tiong, W. K.

MATHEMATICAL FORMULAE BOOK / W. K. TIONG, S. N. SZE & K. L. CHIEW.

ISBN 978-967-2008-11-8

1. Mathematics-Formulae. 2. Mathematic.

1. Sze, S. N. II. Chiew, K. I. II. Judul.

510 ^v

ýýI,

1

ýýýý

^,,

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Pusat Khidmat 1"laivueilat Akademik l? tilý'F: Iiýt't't ýtAl.. Ati"tilA ti; 1RANtiAK

PREFACE

The "Mathematical Formulae Book" is a compilation of useful and impor- tant mathematical formulae, designed specially for undergraduate students.

Detailed explanations of each formula are not included and thus, students should not replace their actual textbook with this book. The mathematical formulae book were selected from various topics in undergraduate mathematics courses available in major textbooks related to:

" Discrete Mathematics

" Linear Algebra

" Statistics

" Calculus

" Multivariable Calculus

" Vector Calculus

" Differential Equations

" Numerical Methods

" Operational Research

Therefore, undergraduate as well as postgraduate students can use this book as a quick reference for formulae at anytime even during examina- tions.

Though this book has been revised several times, there are obviously many rooms for improvement. We welcome any feedbacks, comments and suggestions in the hope to increase the quality of this book.

W. K. Tiong S. N. Sze K. L. Chiew

March 2016

vii

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ALGEBRA

ýGýjý.

ý; ýli - ýýý.

=_C.

_, ýý

u(b+c) = ub+uc, a cad+bc

bd bd

n+c nC

b bb

a

a ýl (I (I _x=-. bý bc ý/

fill = I.

ant

a+n -+t an

'rn , 'rr - 'rrr rn

(am)n

_ amn = ((Irr ym

a Va-)'" a-n =1 _ni'

(ab)' = ab", al/n =Vn.

b ýh.

ý

I

(al n a1!

bI b"

-I bi -- - b"

(a + b) 2= a2 + 2ab + b2, (a - b)2 = a2 - 2ab + b2, (a + b)(a - b) = a2 -b2 ,

(a + b)(a2 - ab + b2) = a3 + b3, (a - b)(a2 + ab + b2) = a3 - bs

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4- b12

+c-

b'-'

- ^2a

-.

^4a

ý MT .

-b f b2

- 4ac If ax` + bx + c= U, then x=

inmTn

2a

For any positive base aý1, the expression log. x=y means ay = x.

Iogr .Z I11ýý

1

a (-ry) lugQ 2n

h";

"Wy)

a ioga i f, (1r

I ýh a

= In x;

= 0;

= 1;

log. x+ log y;

=n log, x;

= log. x- log. y;

= x;

log. x/ log. b, log. x" logt a.

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c

.Q

U.

ß

CL

ý+ "ý ý++

öQ+ö -6 "=

au -+

p üH

+. _.

r^ .ý. "a)

.

ßNýUO

acä+ `` y+

O++öiF.

E _Z m4 +üýa

ýQ+ "y QQwN+

tn aQ zý

mC al ý, C

M (1) ü II G+ý II

. _ý

(A wyu- tp +

CL x

yC+ .

ý. +

Qy`yy

'" + a, _fu"-N

_ID ýö

WAW II

-uü+..

ö

c=. =H

rv fp ir Ný+

ý rv ý

.. p C3 "

ÖGäaHG+

f0 +

t0 f0

.ö c0 H

c yý Fi N LO +Ný.. i .p

.-d""N" Zý

fp y ýi

N

ýNMC

o ýj UUU

a"" ý

ý""

L

H

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TRIGONOMETRY

7r radians = 180°. 1 rad = 1800

10 _ý rad.

7r 180

H radians ýin 0 (. ()"0 t al 10 ()° 0010

300 n/6 1/2 f/2 f/3

15° n/4 f /2 vý2-/2 1

t; 0° 7r/3 f/2 1/2 f

')0° ^7f/2 10

I3

The Law of Sines

sin A sin B sin C

a6c

A

vu

The Law of Cosines

a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C

b

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Pusat Khidmat Maklumat Akademik i; NlX'FR! +! TI NtALA1'SIA iAR N'VAK

.ý. ýý .ý

r= sin x

= cosec x tan x

t' = sec x:

n2 ir ýý

.

n. 2; n

y= cot x

2. n

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Trigonometric Identities

ý, iu(( 11

t, ulo sec a cosec (l -

4 "`i1ý COS (1 sill (1,

OS (1 cot a =-

sin (/

, in(90° - a) = cos a (sin 90° + a).

()s(90° - a) = sina -cos(90° +a).

rui(90° - a) cot a,

, in( -B) - sin B, Cos 0.

tan(-9) __ -tanH,

i +cos2a - 1,

tan 2a= sec2 a,

ý ýý

1- c()t' u c(). ri"'n.

.ý.. ý..

u- b) _- _, ^illui( '11, U, illll.

ýýýý! nfb) = cosucosh: F siuclsiub.

r 1u nfb) -- tan u± tan h I= tali n tan h'

ýý -I IFITTMI

sin 2u ^- 2sillacosa,

cos 2a ⁼ ^Cos2 _a- ^sill ² _a.

=2 cos2 a-1,

=1 -2sill 2a, tan 2a 2taiia

1- ttilll (1

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Sum to Product Formulae

sin a± sire b- 2 sin a2±b) +

cos ab

C2

coca + cos b- 2 cos ab

cos alb,

cosa - nosh -- -2 sin

Ca+b)

^sin

Ca

^b)

,

2 sin a cos b sin(a + b) + sin(a - b).

2 cos a cos b= cos(a - b) + cos(a + b).

2 cos a sin b sin(a + b) - sin(a - b).

2 sin a sin b cos(a - b) - cos(a + b).

tan a tan b cos(a - b) - cos(a + b) -

("()s((/ - h) + ýuti(a + h) .

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HYPERBOLIC FUNCTIONS

ý. ,

e-r 2

. ý"ý ! i. r -

cosh x' sinh z t, uih .r-

cosh x*

e. ⁼ ^{- e-r}

(I ^{+ f-r}

1 tanh2 x=

)rlt '. r -1= ý

-iuh 2x = (,,., h 2x = ý ýsh2 X =

er ⁺ e-. ^r

cosh z

cosech x=

cothx -

2 1

sinh r' 1

tanhr'

1.

sech 2r,

cosech 2r,

2 Binh r cosh r,

cosh2 r+ sinh2 r, cosh 2r +1

2

, inh2 x= cosh 2x -1 2

sinh-1 (ý)

log,,

2a

Cosh-1 -= loge

tanh-1

\n/

. c+ X +a2 a

. Z+ I2-a2

a x>a,

1 l

1ogr (a+ý

IrI < a.

2 a-. ý '

a

sech -lx = cosh-1

1

_cosech _{-1 x=} _Binh-1 ^1, coth-I x= tanh-,

1

xxx

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Graphs of Hyperbolic Functions

Y

y=sinhx

X

r

y=coshx

X

y=tanhx

Yý

y= sech x

-

x

Y

y= cosech x

y= coth. r

X

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GEOMETRY

HIE93LEMMM

Triangle

1-1 hh 1

uL. in H .,.,

Circle

.4= 7rr2

(' ý! -ý

Sector of Circle

1 rH: , s= rH

Sphere

A= -1; rý I=ý -; r3 3

Cylinder

A= 2-r - 27rrh V= irr2h

Cone

r` t h'

V= 3ýrr2h

il

t---I

h

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" The distance between (xi, yl, ^zl) and (x2, y2, z2) is

(xl - x2)2 + (yl - Y2)2 + (zl - z2)2.

" The area of a triangle with vertices (x1, yl ), (x2, Y2) and (X3, Y3) in the (x. y) plane is

1 2

,ý TTf1Iff ,

XI X2 X3

yi y2 Y3 111

General formula : ax + by +c=0.

Slope m and intercept h. y= mx + h.

Intercepts g and h: x+h=1.

9

Slope m and point (xi, yi) y- yi = 111(x -x]).

Two points (x1, yi), (x2iy2) y-Y1

_x- xl

y2-Y1 x2-xl

Angle 0 between two lines tan O= ml - m2 1+ mime

ý

Standard form : (x

- h)2 + (y

- k)2 = r' General form : x2 + y2 + 2gx +2fy+c=0

Radius of a circle r= Vl'f-2 + g2 -c

Parametric form :x=a+r cos t, y=b+r sin t

m

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".. ýý

11)(y - A)

!ik'_ 4p(: r - h) F(h p. k) or F(h. k+p) ý

Iý -h)"

+ (y-1,

a>6

a2 62

F(hfc, k) or F(h, kfc)

ý r-mT

General formula: iii - blu -- (l - n.

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GRAPHS OF COMMON FUNCTIONS

Y}

y= -x

y=X

X

v= -X3

^y=x;;

X

y=logx

X ____. --- ---

Y= -logx

y

y=es

y=-x2

y= -1/x

y=x2

Y= vx

y=-, r

X

y=1/x

-- ---... X

y=e`

x

y=-e-F ýý y--

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Truth Tables

p

T F

/) 9

TT TF

FT FF

,N

F

T

P, ý-9

F T

T F

J) (j

TT TF

FT FF

l) 9

TT TF

FT FF

ýý

T F T

T

Equivalences Laws Name

pAT -- p Identity laws

pvF -p pvT -T Domination laws

pAF F

Idempotent laws pVp p

pAp p

Commutative laws pvq

pAq=

qvp qAp hn9

T F

F F

p -+q

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ie negation iaw ^{11 I}

(p V q) VrpV (q V r) Associative laws

(pAg)ArpA(qAr)

pV (q A r) (p V q) A (p V r) Distributive laws

pA(gV r) (pAg)V (pAr)

De Morgan's laws -(p A q) ýp V -q -(pVq)-, pA-q

pV (p A q) p Absorption laws

pA(pVq)p

Negation laws pV -p = T pA -p F

ý" .... ý

L1I

pi4-'pV9

p-4 9ýQ-+ ýp

pV4ýp--* q

pn4-(p--* ý4) ý(p-+ 9)=pný4)

(pý4)A (p-4 r)p-ý (9A (pr)A (9--* r)(pV4)-*f

(p4)V (p-4 r)p-ý (4Vr) (p ý r) V (4 -4 r) = (p n 4) -+ r

11111 i ^iffill ^{11 111}

3Wj-j; lro M@UHU'TfFoll&lF-lM=nt

pH4(pH4)n(4ýp) pH4-pH-'9

pH4=(pn4)U(-'pn-, Q)

I -(p t-a q) °p 44 -4

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SET

nu

" The universal set is denoted by 1 The empty set or null set is denoted by 0, or { }. The complement of .4 is denoted by . 1` or J.

The union of A and B is denoted by .4UB. The intersection of .4 and B is denoted by

.4nB.

"A ii B means A is a subset of B. Two sets A, B are equal, .4=B if and only if ACBandBCA.

" If .1B and A B, then we write ACB and A is said to be proper subset of A.

ý =77 :ý

N= the set of natural numbers to, 1, '2.3.... }

Z= the set of integers {... , -3, -( 2, -1,0,1,2,3, ... }l Q= the set of rational numbers {Pp, qEZ, q#0}

q JJJ

R= the set of real numbers, i. e. all numbers expressible as finite or infinite decimal expressions

C= the set of complex numbers {. r 1ý ii/ !

. r. ý/ Fß}

For any sets A. B, C and X,

JA uBI = IAI +IBI - JAnBl,

lAuBuCI = IAI+IBI+ICI-lAnBl-lAnCi-lBnCI +JA nBn Cl,

JA x BI = IAJIBI,

IP(X)i = 2" where n= IXI.

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Set Identities

Au0=A

Identity laws

AnU=A AuU=U

Domination laws

An0=0 AUA=A Idempotent laws

AnA=A

AU B= BUA Commutative laws

AnB=BnA

AU (B U C) _ (A U B) UC Associative laws

An (BnC)=(AnB)nC

An(BUC)=(AnB)U(AnC)

Distributive laws AU (B n C) _ (A U B) n (A U C)

' (A U B)° = A° n BC

s laws De Morgan

(A n B)c = Ac u B", AU (A n B) = A

Absorption laws

An (A UB) =A

AUA'=U

Complement laws

An A` =0

(A")"=A

Mathematical Formulae Book (24pgs).pdf

MATHEMATICAL FORMULAE

Contents

HYPERBOLIC FUNCTIONS GEOMETRY

PREFACE

ALGEBRA

TRIGONOMETRY

Sum to Product Formulae

GEOMETRY

GRAPHS OF COMMON FUNCTIONS

JA uBI = IAI +IBI - JAnBl,