radians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side
|
|
- Diana Gibbs
- 5 years ago
- Views:
Transcription
1 . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an angle is said to be in standad position if its vete is at the oigin O and its initial side coincides with the positive ais (Figue.). An angle is said to be in a cetain quadant if, when the angle is in standad position, the teminal side lies in that quadant. Fo instance, a 6 angle lies in quadant I o is simpl said to be a quadant I angle. As Figue. b shows, an angle of measue 8 is a quadant II angle. If the teminal side of an angle in standad position lies along eithe the ais o the ais, then the angle is called quadantal. Fo eample, 60, 0, 80, 90, 0, 90, 80, 0, 60 ae all quadantal angles. Evidentl, an angle is quadantal if and onl if its measue is an intege multiple of 90 ( o adians). Figue. is in standad position Teminal side i Initial side Figue. (a) quadant I angle (b) quadant II angle Teminal side 6 Teminal side 8 8
2 Definition.: Tigonometic Functions of a Geneal Angle Let be an angle in standad position and suppose that (, ) is an point othe than ( 0, 0 ) on the teminal side of (Figue.). If is the distance between (, ) and ( 0, 0 ), then the si tigonometic functions of ae defined b Figue. sin cos tan csc sec cot povided that the denominatos ae not zeo. (, ) O Using simila tiangles, ou can see that the values of the si tigonometic functions in Definition. depend onl on the angle and not on the choice of the point (, ) on the teminal side of. Eample Evaluate the si tigonometic functions of the angle in standad position if the teminal side of contains the point (, ) (, ). Hee,,, and Thus, ( ). sin cos tan csc sec cot. You can detemine the algebaic signs of the tigonometic functions fo angles in the vaious quadants b ecalling the algebaic signs of and in these quadants and 9
3 emembeing that is alwas positive. Fo instance, as Figue. shows, sin is positive in quadants I and II (whee both and ae positive), and it is negative in quadants III and IV (whee is negative and is positive). B poceeding in a simila wa, ou can detemine the signs of the emaining tigonometic functions in the vaious quadants and thus confim the esults in Table.. Figue. > 0 Z > 0 Z sin θ > 0 in Quadant II sin θ > 0 in Quadant I < 0 Z O < 0 Z sin θ < 0 in Quadant III sin θ < 0 in Quadant IV Table. Quadant Containing I II III IV Positive Functions All sin, csc tan, cot cos, sec Negative Functions None cos, sec, tan, cot sin, csc, cos, sec sin, csc, tan, cot Eample Find the quadant in which lies if tan > 0 and sin < 0. This eample can be woked b using Table.; howeve, athe than eling on the table, we pefe to eason as follows: Let (, ) be a point othe than the oigin on the teminal side of (in standad position). Because tan > 0, we see that and have the same algebaic sign. Futhemoe, since sin < 0, it follows that < 0. Because < 0 and < 0, the angle is in quadant III. 0
4 Recipocal Identities If is an angle fo which the functions ae defined, then: (i) csc sin (ii) sec cos (iii) cot tan. Quotient Identities If is an angle fo which the functions ae defined, then: sin cos tan and cot. cos sin Eample If sin and cos, find the values of the othe fou tigonometic functions of. tan sec cot csc sin cos cos tan sin. B using the ecipocal and quotient identities, ou can quickl ecall the algebaic signs of the secant, cosecant, tangent, and cotangent in the fou quadants (Table ), if ou know the algebaic signs of the sine and cosine in these quadants. Anothe impotant identit is deived as follows: Again suppose that is an angle in standad position and that (, ) is a point othe than the oigin on the teminal side of (Figue 9). Because, we have (cos ) (sin ) + +, so + The elationship: (cos ) (sin ) is called the fundamental Pthagoean identit because its deivation involves the fact that +, which is a consequence of the Pthagoean theoem..
5 The fundamental Pthagoean identit is used quite often, and it would be bothesome to wite the paentheses each time fo (cos ) and (sin ) ; et, if the paentheses wee simpl omitted, the esulting epessions would be misundestood. (Fo instance, cos is usuall undestood to mean the cosine of the squae of.) Theefoe, it is customa to wite cos and sin to mean (cos ) and (sin ). Simila notation is used fo the emaining tigonometic functions and fo powes othe than. Thus, cot means ( cot ), n n sec means ( sec ), and so foth. With this notation, the fundamental Pthagoean identit becomes cos + sin. Actuall, thee ae thee Pthagoean identities the fundamental identit and two othes deived fom it. Pthagoean Identities If is an angle fo which the functions ae defined, then: (i) cos + sin (ii) tan sec (iii) cot csc We alead poved (i). To pove (ii), we divide both sides of (i) b + sin cos sin o + cos povided that cos 0. Since sin tan and cos cos cos cos sec,, cos to obtain we have that + tan sec. Identit (iii) is poved b dividing both sides of (i) b sin. Eample The value of one of the tigonometic functions of an angle is given along with the infomation about the quadant in which lies, Find the values of the othe five tigonometic functions of :
6 ( a ) sin, in quadant II. B the fundamental Pthagoean identit, cos sin cos + 69 sin, so Theefoe, cos. 69 Because is in quadant II, we know that cos is negative; hence, cos. It follows that sin tan cos sec cos cot tan csc sin ( b ) tan and sin < 0. Because tan < 0 onl in quadants II and IV, and sin < 0 onl in quadants III and IV, it follows that must be in quadant IV. B pat (ii) sec tan, so sec tan Since is in quadant IV, sec > 0; hence, sec. Because sec it follows that cos sec Now, tan, cos sin cos 8 so sin (tan )(cos ) Finall, csc and cot tan sin
7 In the applications of tigonomet, and especiall in calculus, it is often necessa to make tigonometic calculations, as we have done in this section, without the use of calculatos o tables. Section Poblems In poblems to 0, sketch two coteminal angles and in standad position whose teminal side contains the given point. Aange it so that is positive, is negative, and neithe angle eceeds one evolution. In each case, name the quadant in which the angle lies, o indicate that the angle is quadantal.. (, ). (, ). (, 0 ). (, ). (, ) 6. ( 0, ). (, ) 8. (, 0 ) 9. (, ) 0. ( 0, ) In poblems to 8, specif and sketch thee angles that ae coteminal with the given angle in standad position In poblems 6 to 8, evaluate the si tigonometic functions of the angle in standad position if the teminal side of contains the given point (, ). [Do not use a calculato leave all answes in the fom of a faction o an intege.] In each case, sketch one of the coteminal angles. 9. (, ) 0. (, ). (, ). (, ). (, ). (, ). (, ) 6. (, ). (, ) 8. ( 0, ) 9. Is thee an angle fo which sin? Eplain.
8 0. Using simila tiangles, show that the values of the si tigonometic functions in Definition depend onl on the angle and not on the choice of the point (, ) on the teminal side of.. In each case, assume that is an angle in standad position and find the quadant in which it lies. (a) tan > 0 and sec > 0 (b) sin > 0 and sec < 0 (c) sin > 0 and cos < 0 (d) sec > 0 and tan < 0 (e) tan > 0 and csc < 0 (f) cos < 0 and csc < 0 (g) sec > 0 and cot < 0 (h) cot > 0 and sin > 0. Is thee an angle fo which sin > 0 and csc < 0? Eplain.. Give the algebaic sign of each of the following. (a) cos 6 (b) sin o (c) sec (d) tan 8 (e) cot (f) csc 8 (g) sec. If is an angle fo which the functions ae defined, show that sec ( sin )( tan ) cos.. If sin and cos, use the ecipocal and quotient identities to find (a) sec (b) csc (c) tan (d) cot.
9 6. If sec and csc, use the ecipocal and quotient identities to find (a) sin (b) cos (c) tan (d) cot. In Poblems to 8, the values of one of the tigonometic functions of an angle is given along with infomation about the quadant (Q) in which lies. Find the values of the othe five tigonometic functions of.. sin, in Q I 8. cos, in Q IV 9. sin, in Q III 0. sin, not in Q I. cos, sin < 0. cos, not in Q I. csc, in Q I. sec, in Q III. tan, in Q I 6. tan, sin < 0. cot, csc > 0 8. csc, sec < 0 6
P.7 Trigonometry. What s round and can cause major headaches? The Unit Circle.
P.7 Tigonomet What s ound and can cause majo headaches? The Unit Cicle. The Unit Cicle will onl cause ou headaches if ou don t know it. Using the Unit Cicle in Calculus is equivalent to using ou multiplication
More informationPDF Created with deskpdf PDF Writer - Trial ::
A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationThe 1958 musical Merry Andrew starred Danny Kaye as
The 1958 musical Me Andew staed Dann Kae as Andew Laabee, a teache with a flai fo using unconventional methods in his classes. He uses a musical numbe to teach the Pthagoean theoem, singing and dancing
More information1.6. Trigonometric Functions. 48 Chapter 1: Preliminaries. Radian Measure
48 Chapte : Peliminaies.6 Tigonometic Functions Cicle B' B θ C A Unit of cicle adius FIGURE.63 The adian measue of angle ACB is the length u of ac AB on the unit cicle centeed at C. The value of u can
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # Review Math (Pe -calculus) Name MULTIPLE CHOICE. Choose the one altenative that best completes the statement o answes the question. Use an identit to find the value of the epession. Do not use a
More informationTrigonometric Functions of Any Angle 9.3 (, 3. Essential Question How can you use the unit circle to define the trigonometric functions of any angle?
9. Tigonometic Functions of An Angle Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with, ) a point on the teminal
More informationName Date. Trigonometric Functions of Any Angle For use with Exploration 5.3
5.3 Tigonometic Functions of An Angle Fo use with Eploation 5.3 Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with,
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More information4.3 Right Triangle Trigonometry
Section. Right Tiangle Tigonomet 77. Right Tiangle Tigonomet The Si Tigonometic Functions Ou second look at the tigonometic functions is fom a ight tiangle pespective. Conside a ight tiangle, with one
More informationENGR 1990 Engineering Mathematics Application of Trigonometric Functions in Mechanical Engineering: Part II
ENGR 990 Engineeing Mathematics pplication of Tigonometic Functions in Mechanical Engineeing: Pat II Poblem: Find the coodinates of the end-point of a two-link plana obot am Given: The lengths of the links
More information5.8 Trigonometric Equations
5.8 Tigonometic Equations To calculate the angle at which a cuved section of highwa should be banked, an enginee uses the equation tan =, whee is the angle of the 224 000 bank and v is the speed limit
More informationTrigonometry Standard Position and Radians
MHF 4UI Unit 6 Day 1 Tigonomety Standad Position and Radians A. Standad Position of an Angle teminal am initial am Angle is in standad position when the initial am is the positive x-axis and the vetex
More informationChapter 1: Introduction to Polar Coordinates
Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,
More informationPractice Integration Math 120 Calculus I Fall 2015
Pactice Integation Math 0 Calculus I Fall 05 Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps... ( + d. Hint. Answe. ( 8 t + t + This fist set of indefinite
More informationPractice Integration Math 120 Calculus I D Joyce, Fall 2013
Pactice Integation Math 0 Calculus I D Joyce, Fall 0 This fist set of indefinite integals, that is, antideivatives, only depends on a few pinciples of integation, the fist being that integation is invese
More informationChapter 5: Trigonometric Functions of Angles
Chapte 5: Tigonometic Functions of Angles In the pevious chaptes we have exploed a vaiety of functions which could be combined to fom a vaiety of shapes. In this discussion, one common shape has been missing:
More informationIn many engineering and other applications, the. variable) will often depend on several other quantities (independent variables).
II PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES In man engineeing and othe applications, the behaviou o a cetain quantit dependent vaiable will oten depend on seveal othe quantities independent
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationTransition to College Math
Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain
More information4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationRadian and Degree Measure
CHAT Pe-Calculus Radian and Degee Measue *Tigonomety comes fom the Geek wod meaning measuement of tiangles. It pimaily dealt with angles and tiangles as it petained to navigation, astonomy, and suveying.
More informationB da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
More informationCOORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
More informationINTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
More informationRadian Measure CHAPTER 5 MODELLING PERIODIC FUNCTIONS
5.4 Radian Measue So fa, ou hae measued angles in degees, with 60 being one eolution aound a cicle. Thee is anothe wa to measue angles called adian measue. With adian measue, the ac length of a cicle is
More informationDouble-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities
Double-angle & powe-eduction identities Pat 5, Tigonomety Lectue 5a, Double Angle and Powe Reduction Fomulas In the pevious pesentation we developed fomulas fo cos( β) and sin( β) These fomulas lead natually
More informationCALCULUS II Vectors. Paul Dawkins
CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx
More informationf h = u, h g = v, we have u + v = f g. So, we wish
Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,
More information5.3 Properties of Trigonometric Functions Objectives
Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationRelated Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
More information9.1 POLAR COORDINATES
9. Pola Coodinates Contempoay Calculus 9. POLAR COORDINATES The ectangula coodinate system is immensely useful, but it is not the only way to assign an addess to a point in the plane and sometimes it is
More informationPhysics for Scientists and Engineers
Phsics 111 Sections 003 and 005 Instucto: Pof. Haimin Wang E-mail: haimin@flae.njit.edu Phone: 973-596-5781 Office: 460 Tienan Hall Homepage: http://sola.njit.edu/~haimin Office Hou: 2:30 to 3:50 Monda
More informationMATH 155/GRACEY CH. 10 PRACTICE. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH /GRACEY CH. PRACTICE Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. At the given point, find the line that is nomal to the cuve at the given point.
More information10.1 Angles and their Measure
0. Angles and thei Measue This section begins ou stud of Tigonomet and to get stated, we ecall some basic definitions fom Geomet. A a is usuall descibed as a half-line and can be thought of as a line segment
More informationSolving Some Definite Integrals Using Parseval s Theorem
Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More informationPolar Coordinates. a) (2; 30 ) b) (5; 120 ) c) (6; 270 ) d) (9; 330 ) e) (4; 45 )
Pola Coodinates We now intoduce anothe method of labelling oints in a lane. We stat by xing a oint in the lane. It is called the ole. A standad choice fo the ole is the oigin (0; 0) fo the Catezian coodinate
More informationExercise Set 4.1: Special Right Triangles and Trigonometric Ratios
Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationPHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0
More informationME 210 Applied Mathematics for Mechanical Engineers
Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the
More informationIntroduction and Vectors
SOLUTIONS TO PROBLEMS Intoduction and Vectos Section 1.1 Standads of Length, Mass, and Time *P1.4 Fo eithe sphee the volume is V = 4! and the mass is m =!V =! 4. We divide this equation fo the lage sphee
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationFoundations of Trigonometry
Chapte 0 Foundations of Tigonomet 0. Angles and thei Measue This section begins ou stud of Tigonomet and to get stated, we ecall some basic definitions fom Geomet. A a is usuall descibed as a half-line
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationA Hartree-Fock Example Using Helium
Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty June 6 A Hatee-Fock Example Using Helium Cal W. David Univesity of Connecticut, Cal.David@uconn.edu Follow
More informationPHYS 301 HOMEWORK #10 (Optional HW)
PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More information0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS Intenational Geneal Cetificate of Seconday Education MARK SCHEME fo the Octobe/Novembe 009 question pape fo the guidance of teaches 0606 ADDITIONAL MATHEMATICS
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationREVIEW Polar Coordinates and Equations
REVIEW 9.1-9.4 Pola Coodinates and Equations You ae familia with plotting with a ectangula coodinate system. We ae going to look at a new coodinate system called the pola coodinate system. The cente of
More informationPhysics 521. Math Review SCIENTIFIC NOTATION SIGNIFICANT FIGURES. Rules for Significant Figures
Physics 51 Math Review SCIENIFIC NOAION Scientific Notation is based on exponential notation (whee decimal places ae expessed as a powe of 10). he numeical pat of the measuement is expessed as a numbe
More informationtransformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface
. CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),
More informationRECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationto point uphill and to be equal to its maximum value, in which case f s, max = μsfn
Chapte 6 16. (a) In this situation, we take f s to point uphill and to be equal to its maximum value, in which case f s, max = μsf applies, whee μ s = 0.5. pplying ewton s second law to the block of mass
More informationChapter 1. Introduction
Chapte 1 Intoduction 1.1 The Natue of Phsics Phsics has developed out of the effots of men and women to eplain ou phsical envionment. Phsics encompasses a emakable vaiet of phenomena: planeta obits adio
More informationΔt The textbook chooses to say that the average velocity is
1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the
More informationPhysics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving
Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act,
More informationSolutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook
Solutions to Poblems Chapte 9 Poblems appeae on the en of chapte 9 of the Textbook 8. Pictue the Poblem Two point chages exet an electostatic foce on each othe. Stategy Solve Coulomb s law (equation 9-5)
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More information11.2 Proving Figures are Similar Using Transformations
Name lass ate 11. Poving igues ae Simila Using Tansfomations ssential Question: How can similait tansfomations be used to show two figues ae simila? esouce ocke ploe onfiming Similait similait tansfomation
More informationVectors, Vector Calculus, and Coordinate Systems
Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any
More informationChapter 2: Introduction to Implicit Equations
Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationPhys 201A. Homework 5 Solutions
Phys 201A Homewok 5 Solutions 3. In each of the thee cases, you can find the changes in the velocity vectos by adding the second vecto to the additive invese of the fist and dawing the esultant, and by
More informationMuch that has already been said about changes of variable relates to transformations between different coordinate systems.
MULTIPLE INTEGRLS I P Calculus Cooinate Sstems Much that has alea been sai about changes of vaiable elates to tansfomations between iffeent cooinate sstems. The main cooinate sstems use in the solution
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationElectric Charge and Field
lectic Chage and ield Chapte 6 (Giancoli) All sections ecept 6.0 (Gauss s law) Compaison between the lectic and the Gavitational foces Both have long ange, The electic chage of an object plas the same
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug 4 Complete Section Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a
More informationESCI 342 Atmospheric Dynamics I Lesson 3 Fundamental Forces II
Reading: Matin, Section. ROTATING REFERENCE FRAMES ESCI 34 Atmospheic Dnamics I Lesson 3 Fundamental Foces II A efeence fame in which an object with zeo net foce on it does not acceleate is known as an
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationApplication of Parseval s Theorem on Evaluating Some Definite Integrals
Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at http://pubs.sciepub.com/tjant/// Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating
More informationChapter 4 Trigonometric Functions
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math Pecalculus Ch. 6 Review Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Solve the tiangle. ) ) 6 7 0 Two sides and an angle (SSA) of a tiangle ae
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Chapte 7-8 Review Math 1316 Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Solve the tiangle. 1) B = 34.4 C = 114.2 b = 29.0 1) Solve the poblem. 2) Two
More information4.3 Area of a Sector. Area of a Sector Section
ea of a Secto Section 4. 9 4. ea of a Secto In geomety you leaned that the aea of a cicle of adius is π 2. We will now lean how to find the aea of a secto of a cicle. secto is the egion bounded by a cental
More informationBASIC ALGEBRA OF VECTORS
Fomulae Fo u Vecto Algeba By Mi Mohammed Abbas II PCMB 'A' Impotant Tems, Definitions & Fomulae 01 Vecto - Basic Intoduction: A quantity having magnitude as well as the diection is called vecto It is denoted
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More informationPart V: Closed-form solutions to Loop Closure Equations
Pat V: Closed-fom solutions to Loop Closue Equations This section will eview the closed-fom solutions techniques fo loop closue equations. The following thee cases will be consideed. ) Two unknown angles
More informationdq 1 (5) q 1 where the previously mentioned limit has been taken.
1 Vecto Calculus And Continuum Consevation Equations In Cuvilinea Othogonal Coodinates Robet Maska: Novembe 25, 2008 In ode to ewite the consevation equations(continuit, momentum, eneg) to some cuvilinea
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationMath Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
More information1. Show that the volume of the solid shown can be represented by the polynomial 6x x.
7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends
More information