This document is a mathematical handbook that provides a collection of important mathematical formulas and tables for use by students and researchers across various fields including mathematics, physics, engineering, and other sciences. It contains formulas ranging from elementary to advanced topics and is divided into two main parts, with the first part presenting formulas and explanatory material and the second part containing numerical tables. The goal is to provide a single volume reference for commonly used mathematical results.
How use Modelica? Read this note for better understanding. Professionally written and explained. Good for software development by beginners in Modelica .
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Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
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C++ programming language notes for beginners and Collage students. Written for beginners. Colored graphics. Function by Function explanation with complete examples. Well commented examples. Illustrations are made available for data dealing at memory level.
How use Modelica? Read this note for better understanding. Professionally written and explained. Good for software development by beginners in Modelica .
Basic ForTran Programming - for Beginners - An Introduction by Arun Umraossuserd6b1fd
ForTran Programming Notes for Beginners. With complete example. Covers full syllabus. Well structured and commented for easy - self understanding. Good for Senior Secondary Students.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
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C++ programming language notes for beginners and Collage students. Written for beginners. Colored graphics. Function by Function explanation with complete examples. Well commented examples. Illustrations are made available for data dealing at memory level.
Welding Defects
Eurotech Now inteducing Welding Defects. Welding Defect is any type of flaw in the object which requires welding. Seven type of Welding Defect
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Introduction to Abstract Algebra by
D. S. Malik
Creighton University
John N. Mordeson
Creighton University
M.K. Sen
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It includes the most important sections of abstract mathematics like Sets, Relations, Integers, Groups, Permutation Groups, Subgroups and Normal Subgroups, Homomorphisms and Isomorphisms of Groups, Rings etc.
This book is grate not only for those who study in mathematics departments, but for all who want to start with abstract mathematics. Abstract mathematics is very important for computer sciences and engineering.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
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• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
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Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
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Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
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- A fully editable and extendable library for grid component modelling;
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Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
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Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
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Epistemic Interaction - tuning interfaces to provide information for AI support
Mathematical handbook of formulas and tables manteshwer
1. P r e f
The pur-pose of this handbook is to supply a collection of mathematical formulas and
tables which will prove to be valuable to students and research workers in the fields of
mathematics, physics, engineering and other sciences. TO accomplish this, tare has been
taken to include those formulas and tables which are most likely to be needed in practice
rather than highly specialized results which are rarely used. Every effort has been made
to present results concisely as well as precisely SOthat they may be referred to with a maxi-
mum of ease as well as confidence.
Topics covered range from elementary to advanced. Elementary topics include those
from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics
include those from differential equations, vector analysis, Fourier series, gamma and beta
functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions
and various other special functions of importance. This wide coverage of topics has been
adopted SOas to provide within a single volume most of the important mathematical results
needed by the student or research worker regardless of his particular field of interest or
level of attainment.
The book is divided into two main parts. Part 1 presents mathematical formulas
together with other material, such as definitions, theorems, graphs, diagrams, etc., essential
for proper understanding and application of the formulas. Included in this first part are
extensive tables of integrals and Laplace transforms which should be extremely useful to
the student and research worker. Part II presents numerical tables such as the values of
elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as
advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion,
especially to the beginner in mathematics, the numerical tables for each function are sep-
arated, Thus, for example, the sine and cosine functions for angles in degrees and minutes
are given in separate tables rather than in one table SOthat there is no need to be concerned
about the possibility of errer due to looking in the wrong column or row.
1 wish to thank the various authors and publishers who gave me permission to adapt
data from their books for use in several tables of this handbook. Appropriate references
to such sources are given next to the corresponding tables. In particular 1 am indebted to
the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S.,
and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their
book S T tf B a Ao
a i ba g y
t
M o R ln ie r l e ed sd i o s s t i
c
1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin
for their excellent editorial cooperation.
M. R. SPIEGEL
Rensselaer Polytechnic Institute
September, 1968
2.
3. CONTENTS
Page
1. Special Constants.. ............................................................. 1
2. Special Products and Factors .................................................... 2
3. The Binomial Formula and Binomial Coefficients ................................. 3
4. Geometric Formulas ............................................................ 5
5. Trigonometric Functions ........................................................ 11
6. Complex Numbers ............................................................... 21
7. Exponential and Logarithmic Functions ......................................... 23
8. Hyperbolic Functions ........................................................... 26
9. Solutions of Algebraic Equations ................................................ 32
10. Formulas from Plane Analytic Geometry ........................................ 34
11. Special Plane Curves........~ ................................................... 40
12. Formulas from Solid Analytic Geometry ........................................ 46
13. Derivatives ..................................................................... 53
14. Indefinite Integrals .............................................................. 57
15. Definite Integrals ................................................................ 94
16. The Gamma Function ......................................................... ..10 1
17. The Beta Function ............................................................ ..lO 3
18. Basic Differential Equations and Solutions ..................................... .104
19. Series of Constants..............................................................lO 7
20. Taylor Series...................................................................ll 0
21. Bernoulliand Euler Numbers ................................................. ..114
22. Formulas from Vector Analysis.. ............................................. ..116
23. Fourier Series ................................................................ ..~3 1
24. Bessel Functions.. ............................................................ ..13 6
2s. Legendre Functions.............................................................l4 6
26. Associated Legendre Functions ................................................. .149
27. Hermite Polynomials............................................................l5 1
28. Laguerre Polynomials .......................................................... .153
29. Associated Laguerre Polynomials ................................................ KG
30. Chebyshev Polynomials..........................................................l5 7
7. THE GREEK ALPHABET
Greek Greek tter
Greek
name G&W name Lower case Capital
Alpha A Nu N
Beta B Xi sz
Gamma l? Omicron 0
Delta A Pi IT
Epsilon E Rho P
Zeta Z Sigma 2
Eta H Tau T
Theta (3 Upsilon k
Iota 1 Phi @
Kappa K Chi X
Lambda A Psi *
MU M Omega n
11. 4 THE BINOMIAL FORMULA AND BINOMIAL COElFI?ICIFJNTS
PROPERTIES OF BINOMIAL COEFFiClEblTS
3.6
This leads to Paseal’s triangk [sec page 2361.
3.7 (1) + (y) + (;) + ... + (1) = 27l
3.8 (1) - (y) + (;) - ..+-w(;) = 0
3.9
3.10 (;) + (;) + (7) + .*. = 2n-1
3.11 (y) + (;) + (i) + ..* = 2n-1
3.12
3.13
3.14
-d
MUlTlNOMlAk FORfvlUlA
3.16 (zI+%~+...+zp)~ = ~~~!~~~~~..~~!~~1~~2...~~~
where the mm, denoted by 2, is taken over a11 nonnegative integers % %, . . , np fox- whkh
q+n2+ ... +np = 72..
12. 1
4 GEUMElRlC FORMULAS
&
RECTANGLE OF LENGTH b AND WIDTH a
4.1 Area = ab
4.2 Perimeter = 2a + 2b
b
Fig. 4-1
PARAllELOGRAM OF ALTITUDE h AND BASE b
4.3 Area = bh = ab sin e
4.4 Perimeter = 2a + 2b
1
Fig. 4-2
‘fRlAMf3i.E OF ALTITUDE h AND BASE b
4.5 Area = +bh = +ab sine
ZZZ
I/S(S - a)(s - b)(s - c)
*
where s = &(a + b + c) = semiperimeter
b
4.6 Perimeter = u+ b+ c Fig. 4-3
L,“Z n_
., :
.,, ‘fRAPB%XD C?F At.TlTUDE fz AND PARAl.lEL SlDES u AND b
4.7 Area = 3h(a + b)
4.8 Perimeter = a + b + h Y&+2 /c-
C sin 4
= a + b + h(csc e + csc $)
1
Fig. 4-4
5
/
-
13. 6 GEOMETRIC FORMULAS
REGUkAR POLYGON OF n SIDES EACH CJf 1ENGTH b
COS(AL)
4.9 Area = $nb?- cet c = inbz-
sin (~4%)
4.10 Perimeter = nb
7,’
Fig. 4-5
0.’
CIRÇLE OF RADIUS r
0
4.11 Area = &
4.12 Perimeter = 277r
Fig. 4-6
SEClOR OF CIRCLE OF RAD+US Y
4.13 Area = &r% [e in radians]
T 8
4.14 Arc length s = ~6
0
A
T
Fig. 4-7
RADIUS OF C1RCJ.E INSCRWED tN A TRtANGlE OF SIDES a,b,c
*
&$.s- U)(S Y b)(s -.q)
4.15 r=
s
where s = +(u + b + c) = semiperimeter
Fig. 4-6
RADIUS- OF CtRClE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c
abc
4.16 R=
4ds(s - a)@ - b)(s - c)
where e = -&(a. b + c) = semiperimeter
+
Fig. 4-9
14. G FE OO RM ME
7 UT
3 6 0 °
4 A .
= & sr s 1
= n
+ ise n 7 r
n ni a 2
r n 2
4 P . = 2e s 1 = 2 nr s i y 8 n ri i n r mn z e t e
Fig. 4-10
4 A .
= n t rZ =1 n r t a e T
L 9 r2 an a
! 2 n T ! ! ?
n n I T
:
4 P . = 2e t 2 = 2 nr t a 0 n ri a n r m nk e? t e
0
F 4 i - g 1
SRdMMHW W C%Ct& OF RADWS T
4 A o .s pr f=2 h + ( -a s
e e) 1 a r e r i
a d2 tn e d
e
T r
tz!?
Fig. 4-12
4 A =
. r r 2 a e 2 b a
7r/2
4 P . = e
4a 5 2 4 1 - kz r
s e c3 ii l m
+ @ e t e
0
= 27r@sTq [ a p p r o
w h
k = ~/=/a. See p e254 f n a r to u g e ar m e b F e
4 l i -r e g i
1
4 A =
. $ab r 2 e 4 a
4 a + @ T
4 A l . ABC r = e -&dw
2 c +n E5 gl tn h
1 ) AOC
b
Fig. 4-14
- f
15. 8 GEOMETRIC FORMULAS
RECTANGULAR PARALLELEPIPED OF LENGTH u, HEIGHT r?, WIDTH c
4.26 Volume = ubc
4.27 Surface area = Z(ab + CLC bc)
+
a
Fig. 4-15
PARALLELEPIPED OF CROSS-SECTIONAL AREA A AND HEIGHT h
4.28 Volume = Ah = abcsine
Fig. 4-16
SPHERE OF RADIUS ,r
4.29 Volume = +
1
,------- ---x .
4.30 Surface area = 4wz
@
Fig. 4-17
RIGHT CIRCULAR CYLINDER OF RADIUS T AND HEIGHT h
4.31 Volume = 77&2
h
4.32 Lateral surface area = 25dz
Fig. 4-18
CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT 2
4.33 Volume = m2h = ~41 sine
2wh
4.34 Lateral surface area = 2777-1 = z = 2wh csc e
Fig. 4-19
16. GEOMETRIC FORMULAS 9
CYLINDER OF CROSS-SECTIONAL AREA A AND SLANT HEIGHT I
4.35 Volume = Ah = Alsine
4.36 Lateral surface area = pZ = GPh -
- ph csc t
Note that formulas 4.31 to 4.34 are special cases.
Fig. 4-20
RIGHT CIRCULAR CONE OF RADIUS ,r AND HEIGHT h
4.37 Volume = jîw2/z
4.38 Lateral surface area = 77rd77-D = ~-7-1
Fig. 4-21
PYRAMID OF BASE AREA A AND HEIGHT h
4.39 Volume = +Ah
Fig. 4-22
SPHERICAL CAP OF RADIUS ,r AND HEIGHT h
4.40 Volume (shaded in figure) = &rIt2(3v - h)
4.41 Surface area = 2wh
Fig. 4-23
FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII u,h AND HEIGHT h
4.42 Volume = +h(d + ab + b2)
4.43 Lateral surface area = T(U + b) dF + (b - CL)~
= n(a+b)l
Fig. 4-24
17. 10 GEOMETRIC FORMULAS
SPHEMCAt hiiWW OF ANG%ES A,&C Ubl SPHERE OF RADIUS Y
4.44 Area of triangle ABC = (A + B + C - z-)+
Fig. 4-25
TOW$ &F lNN8R RADlU5 a AND OUTER RADIUS b
4.45 Volume = &z-~(u+ b)(b - u)~
w
4.46 Surface area = 7r2(b2 u2)
-
4.47 Volume = $abc
Fig. 4-27
PARAWlO~D aF REVOllJTlON
T.
4.4a Volume = &bza
Fig. 4-28
18. 5 TRtGOhiOAMTRiC WNCTIONS
D OE T FF R F l I FU A R N G T N
O I l O R C
R G T N I T
Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. The trigonometric functions of
angle A are defined as follows.
opposite B
5 sintzof A
. = sin A 1
= : =
hypotenuse
adjacent
5 cosineof
. A = ~OSA 2
= i =
hypotenuse
opposite
5 tangent . of A = tanA 3 f = -~
=
adjacent
adjacent
5 c . of A = o cet A 4
= k = t c z n g
opposite A
hypotenuse
5.5 secant of A = sec A = t = -~
adjacent
hypotenuse Fig. 5-1
5 cosecant . of A = csc A 6
= z =
opposite
E TX A WOT N M 3 HG
E G A TE I R 9 L Y
N H C E0 E
S A H A’
I
Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates
(%,y) where x is eonsidered as positive along OX and negative along OX’ while y is positive along OY and
negative along OY’. The distance from origin 0 to point P is positive and denoted by r = dm.
The angle A described cozmtwcZockwLse from OX is considered pos&ve. If it is described dockhse from
OX it is considered negathe. We cal1 X’OX and Y’OY the x and y axis respectively.
The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quad-
rants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A
is in the third quadrant.
Y Y
II 1 II 1
III IV III IV
Y’ Y’
Fig. 5-2 Fig. 5-3
11
f
19. 12 TRIGONOMETRIC FUNCTIONS
For an angle A in any quadrant the trigonometric functions of A are defined as follows.
5.7 sin A = ylr
5.8 COS
A = xl?.
5.9 tan A = ylx
5.10 cet A = xly
5.11 sec A = v-lx
5.12 csc A = riy
RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS
N
A radian is that angle e subtended at tenter 0 of a eircle by an arc
MN equal to the radius r. 1 r
Since 2~ radians = 360° we have e
M
0 r
5.13 1 radian = 180°/~ = 57.29577 95130 8232. . . o
5.14 10 = ~/180 radians = 0.01745 32925 19943 29576 92.. .radians B
Fig. 5-4
REkATlONSHlPS AMONG TRtGONOMETRK FUNCTItB4S
5.15 tanA = 5 5.19 sine A + ~OS~A = 1
~II ~ 1
COS A
5.16 &A zz - 5.20 sec2A - tane A = 1
tan A sin A
1
5.17 sec A = ~ 5.21 csce
A - cots A = 1
COS
A
5.18 1
cscA = -
sin A
SIaNS AND VARIATIONS OF TRl@ONOMETRK FUNCTIONS
+ + + + + +
1
0 to 1 1 to 0 0 to m CC 0
to 1 to uz m to 1
II + - - +
1 to 0 0 to -1 -mtoo oto-m -cc to -1 1 to ca
- + +
III
0 to -1 -1 to 0 0 to d Ccto 0 -1to-m --COto-1
- + - + -
IV
-1 to 0 0 to 1 -- too oto-m uz to 1 -1 to --
20. TRIGONOMETRIC FUNCTIONS 1 3
E V X F A
T A O R
L FC R I
U OT
U V G
E F
N
A A O
S C
N R N T
G I
Angle A Angle A
sin A COSA tan A cet A sec A csc A
in degrees in radians
00 0 0 1 0 w 1 cc
15O rIIl2 #-fi) &(&+fi) 2-fi 2+* fi-fi &+fi
300 ii/6 1 +ti *fi fi $fi 2
450 zl4 J-fi $fi 1 1 fi fi
60° VI3 Jti r
1 fi .+fi 2 ;G
750 5~112 i(fi+m @-fi) 2+& 2-& &+fi fi-fi
900 z.12 1 0 *CU 0 km 1
105O 7~112 *(fi+&) -&(&-Y% -(2+fi) -(2-&) -(&+fi) fi-fi
120° 2~13 *fi -* -fi -$fi -2 ++
1350 3714 +fi -*fi -1 -1 -fi h
150° 5~16 4 -+ti -*fi -fi -+fi 2
165O llrll2 $(fi- fi) -&(G+ fi) -(2-fi) -(2+fi) -(fi-fi) Vz+V-c?
180° ?r 0 -1 0 Tm -1 *ca
1950 13~112 -$(fi-fi) -*(&+fi) 2-fi 2 + ti -(&-fi) -(&+fi)
210° 7716 1 - 4 & 6 l f 3 i - g -2 f i
225O 5z-14 -Jfi -*fi 1 1 -fi -fi
240° 4%J3 -# -4 ti &fi -2 -36
255O 17~112 -&&+&Q -&(&-fi) 2+fi 2-6 -(&+?cz) -(fi-fi)
270° 3712 -1 0 km 0 Tm -1
285O 19?rll2 -&(&+fi) *(&-fi) -(2+6) -@-fi) &+fi -(fi-fi)
3000 5ïrl3 -*fi 2 -ti -*fi 2 -$fi
315O 7?rl4 -4fi *fi -1 -1 fi -fi
330° 117rl6 1 *fi -+ti -ti $fi -2
345O 237112 -i(fi- 6) &(&+ fi) -(2 - fi) -(2+6) fi-fi -(&+fi)
360° 2r 0 1 0 T-J 1 ?m
For tables involving other angles see pages 206-211 and 212-215.
f
21.
22.
23.
24.
25.
26. TRIGONOMETRIC FUNCTIONS 19
5.89 y = cet-1% 5.90 y = sec-l% 5.91 y = csc-lx
I
Y
_--/
T /A--
/’
/
,
--- -77
--
//
,
Fig. 5-14 Fig. 5-15 Fig. 5-16
RElAilONSHfPS BETWEEN SIDES AND ANGtGS OY A PkAtM TRlAF4GlG ’
The following results hold for any plane triangle ABC with
sides a, b, c and angles A, B, C. A
5.92 Law of Sines
a b c 1
-=Y=-
sin A sin B sin C
C
5.93 Law of Cosines /A f
cs = a2 + bz - Zab COS C
with similar relations involving the other sides and angles.
5.94 Law of Tangents
a+b tan $(A + B)
-
a-b = tan i(A -B)
with similar relations involving the other sides and angles. Fig. 5-1’7
5.95 sinA = :ds(s - a)(s - b)(s - c)
where s = &a + b + c) is the semiperimeter of the triangle. Similar relations involving angles
B and C cari be obtained.
See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
Spherieal triangle ABC is on the surface of a sphere as shown
in Fig. 5-18. Sides a, b, c [which are arcs of great circles] are
measured by their angles subtended at tenter 0 of the sphere. A, B, C
are the angles opposite sides a, b, c respectively. Then the following
results hold.
5.96 Law of Sines
sin a
-z-x_ sin b sin c
sin A sin B sin C
5.97 Law of Cosines
cosa = cosbcosc + sinbsinccosA
COSA = - COSB COSC + sinB sinccosa
with similar results involving other sides and angles.
27. 2 0 T R F I U G N O C N T O
5 L o
. T a f
9 a w 8 n g e n t s
t & + B a t
( $ ) + n
b a
A ( ) n u
t & - B a= t
( ) n a
A n i (
w s ri i i et o mn s s h t ai v i
a un h n o d
l l d e a l e
g t r rv s
l s e i
5 . 9 9
w s = & h 1+ c S
( e r) uh r
i f e. o m ose
+ o s t a i li
a r u h n l dd
n l e ga e
d t r lr s
5 . 1 0 0
w S = + h+ B + C ( Se r) A i rh f e. o meos o s t a i li
a r u h n l dd
n ld e ga e t r lr
S a f e 4l o
p 1
e .s r
a 0 4o m
g . 4 e
u , l a
NAPIER’S RlJlES FGR RtGHT ANGLED SPHERICAL TRIANGLES
E f r x a o C it c na r , gh e g s i
f p o
r ha p t l
e e
f pv Ar
t tr e
w h i
e Ze
at ih e f t rs o a i
i 3 n
r h r r nc
C
a g i F s ii wn i b -a b i
v c g e le, , u , . .
B 9n A l , d
a
C O
F 5 i - g 1 . 9 F 5 i - g 2 .
S t uq h a pu a e ri p a r c
a s a e io nr F
n i 5 s n ta i
e w
s r w - a e i ng t
h c e2 t
p e C .l h
t g 0 t
r r Oe e
i e
[ c i t
o hn m oc a d
y a p A ai
n
p B l
n nc
do gc
. da
t l
m et ee n n s
i
A o o t n p n o t y a e
f h c fi hc
e r a ms i
i a prt ti tl
s an
cs hd wl pl
ve ea
d ec
o ae
-i r
l da frg
a p da t a
t r n
j h wp
r e d a e c to a oc r
a m a ps r p ea Te lN
a t
p xi hr r l aao s tn eu t e pr
s i nl
s d ie
i
5 T s. o a h m i1 f
p n ee i n0 t a p y q d e 1 ht r rt
o u d f oh t o a a l
e a f eph d d l e
n ae u j s
g r c a
e
5.102 T s o a h m i f n e i n
p e t a p y q d eo ht r rc u d f eh t oo
o t o a l f pe
h dp
s l e ae up s
i r no
c
E S x C = 9
i a- O C n= m
0 9A -- O w c h 0,
° p B
A - e e a °l , B ve e:
s a = t b it ( a na C
o n s n = rt
O i -a n Bn a )b
s ( = C
i C C
a ( n
O OO C ~
o A =-r S OaO
S C s B A O - S
i ) SB n
T c o ch b
a o f oe a e fb
r uts r i rt 5 rh o e p
l e s oa 1. s e n o a
s m 9 e
i 9 ug n .7 l e e
28. A complex number is generally written as a + bi where a and b are real numbers and i, called the
imaginaru unit, has the property that is = -1. The real numbers a and b are called the real and ima&am
parts of a + bi respectively.
The complex numbers a + bi and a - bi are called complex conjugates of each other.
6.1 a+bi = c+di if and only if a=c and b=cZ
6.2 (a + bi) + (c + o!i) = (a + c) + (b + d)i
6.3 (a + bi) - (c + di) = (a - c) + (b - d)i
6.4 (a+ bi)(c+ di) = (ac- bd) + (ad+ bc)i
Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by
-1 wherever it occurs.
21
29. 22 COMPLEX NUMBERS
GRAPH OF A COMPLEX NtJtWtER
A complex number a + bi cari be plotted as a point (a, b) on an
p,----. y
xy plane called an Argand diagram or Gaussian plane. For example
in Fig. 6-1 P represents the complex number -3 + 4i.
A eomplex number cari also be interpreted as a wector from
0 to P.
- X
0
*
Fig. 6-1
POLAR FORM OF A COMPt.EX NUMRER
In Fig. 6-2 point P with coordinates (x, y) represents the complex
number x + iy. Point P cari also be represented by polar coordinates
(r, e). Since x = r COS6, y = r sine we have
6.6 x + iy = ~(COS +
0 i sin 0)
L - X
called the poZar form of the complex number. We often cal1 r = dm
the mocklus and t the amplitude of x + iy.
Fig. 6-2
tWJLltFltCATt43N AND DtVlStON OF CWAPMX NUMBRRS 1bJ POLAR FtMM ilj
0”
6.7 [rl(cos el + i sin ei)] [re(cos ez + i sin es)] = rrrs[cos tel + e2) + i sin tel + e2)]
V-~(COS + i sin el)
e1
6.8 ZZZ 2 [COS(el - e._J + i sin (el - .9&]
rs(cos ee + i sin ez)
DE f#OtVRtt’S THEORRM
If p is any real number, De Moivre’s theorem states that
6.9 [r(cos e + i sin e)]p = rp(cos pe + i sin pe)
. ”
RCWTS OF CfMMWtX NUtMB#RS
If p = l/n where n is any positive integer, 6.9 cari be written
1
e + 2k,, e + 2kH
6.10 [r(cos e + i sin e)]l’n = rl’n ~OS- + i sin ~
L n n
where k is any integer. From this the n nth roots of a complex number cari be obtained by putting
k=O,l,2 ,..., n-l.
30. In the following p, q are real numbers, CL, are positive numbers and WL,~are positive integers.
t
7.1 cp*aq z aP+q 7.2 aP/aqE @-Q 7.3 (&y E rp4
7.4 u”=l, a#0 7.5 a-p = l/ap 7.6 (ab)p = &‘bp
7.7 & z aIIn 7.8 G = pin 7.9 Gb =%Iî/%
In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function y = ax
is called an exponentd function.
If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm of N to the base a. The number
N = ap is called t,he antdogatithm of p to the base a, written arkilogap.
Example: Since 3s = 9 we have log3 9 = 2, antilog3 2 = 9.
The fumAion v = loga x is called a logarithmic jwzction.
7.10 logaMN = loga M + loga N
7.11 log,z ; = logG M - loga N
7.12 loga Mp = p lO& M
Common logarithms and antilogarithms [also called Z?rigg.sian] are those in which the base a = 10.
The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and
antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196.
23
31. 24 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
NATURAL LOGARITHMS AND ANTILOGARITHMS
Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e =
2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural
logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z]
see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF lO@ARlTHMS
The relationship between logarithms of a number N to different bases a and b is given by
hb iv
7.13 loga N = -
hb a
In particular,
7.14 loge N = ln N = 2.30258 50929 94.. . logio N
7.15 logIO = logN
N = 0.43429 44819 03.. . h& N
RElATlONSHlP BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC FUNCT#ONS ;;
7.16 eie = COS + i sin 8,
0 e-iO = COS 13 - i sin 6
These are called Euler’s dent&es. Here i is the imaginary unit [see page 211.
eie e-ie
-
7.17 sine =
2i
eie e-ie
+
7.18 case = 2
7.19
7.20
2
7.21 sec 0 = &O + e-ie
2i
7.22 csc 6 = eie - e-if3
7.23 eiCO+2k~l = eie k = integer
From this it is seen that @ has period 2G.
32. E XA L PN OF OD GU N A 25
N E RC N
POiAR FORfvl OF COMPLEX NUMBERS EXPRESSE$3 AS AN EXPONENTNAL
T p f o h a co o n fe ol + i r c u b w m
x a y ma tm e
i o re
r p n re b [f lx 6
i pi r e 2 s a ep .
t a mr 2 e s x o6
t g
7 . 2 4 6
x + iy = ~(COS + i sin 0) = 9-ei0
OPERATIONS WITH COMPLEX ffUMBERS IN POLAR FORM
F 6 t o 6 .o h r 2
p a
. en r m 2 t 1 t
7 a r qf og u o 0 h uo u
e e l e il g a vl h s a
o
7.27 (q-eio)P q-P&mJ [
zz M t D o h e i e v o r
7.2B (reiO)l/n E [~&O+Zk~~]l/n = rl/neiCO+Zkr)/n
LOGARITHM OF A COMPLEX NUMBER
7.29 l ( = l r n i + 2
+ T n k e= i k
e @n z ) t - e i
33. DEIWWOPI OF HYPRRWLK FUNCTIONS .:‘.C,
# - e-z
8.1 Hyperbolic sine of x = sinh x =
2
ez + e-=
8.2 Hyperbolic cosine of x = coshx =
2
8.3 Hyperbolic tangent of x = tanhx = ~~~~~~
ex + eCz
8.4 Hyperbolic cotangent of x = coth x = es _ e_~
2
8.5 Hyperbolic secant of x = sech x =
ez + eëz
8.6 Hyperbolic cosecant of x = csch x = &
RELATWNSHIPS AMONG HYPERROLIC FUWTIONS
sinh x
8.7 tanhx = a
1 cash x
coth z = - = -
tanh x sinh x
1
sech x = -
cash x
1
8.10 cschx = -
sinh x
8.11 coshsx - sinhzx = 1
8.12 sechzx + tanhzx = 1
8.13 cothzx - cschzx = 1
FUNCTIONS OF NRGA’fWE ARGUMENTS
8.14 sinh (-x) = - sinh x 8.15 cash (-x) = cash x 8.16 tanh (-x) = - tanhx
8.17 csch (-x) = -cschx 8.18 sech(-x) = sechx 8.19 coth (-x) = -~OUIS
26
34. HYPERBOLIC FUNCTIONS 27
AWMWM FORMWAS
0.2Q sinh (x * y) = sinh x coshg * cash x sinh y
8.21 cash (x 2 g) = cash z cash y * sinh x sinh y
8.22 tanhx f tanhg
tanh(x*v) =
12 tanhx tanhg
8.23 coth z coth y 2 1
coth (x * y) =
coth y * coth x
8.24 sinh 2x = 2 ainh x cash x
8.25 cash 2x = coshz x + sinht x = 2 cosh2 x - 1 = 1 + 2 sinh2 z
2 tanh x
8.26 tanh2x =
1 + tanh2 x
HAkF ABJGLR FORMULAS
8.27 sinht = [+ if x > 0, - if x < O]
cash x + 1
8.28 CoshE = -~
2 2
cash x - 1
8.29 tanh; = k [+ if x > 0, - if x < 0]
cash x + 1
Z sinh x ZZ cash x - 1
cash x + 1 sinh x
.4 ’ MUlTWlE A!Wlfi WRMULAS
8.30 sinh 3x = 3 sinh x + 4 sinh3 x
8.31 cosh3x = 4 cosh3 x - 3 cash x
3 tanh x + tanh3 x
8.32 tanh3x =
1 + 3 tanhzx
8.33 sinh 4x = 8 sinh3 x cash x + 4 sinh x cash x
8.34 cash 4x = 8 coshd x - 8 cosh2 x -t- 1
4 tanh x + 4 tanh3 x
8.35 tanh4x =
1 + 6 tanh2 x + tanh4 x