Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
|
|
- Cecilia Davis
- 5 years ago
- Views:
Transcription
1
2 Person Edution Limited Edinurgh Gte Hrlow Essex M20 2JE Englnd nd ssoited ompnies throughout the world Visit us on the World Wide We t: Person Edution Limited 2014 ll rights reserved. No prt of this pulition my e reprodued, stored in retrievl system, or trnsmitted in ny form or y ny mens, eletroni, mehnil, photoopying, reording or otherwise, without either the prior written permission of the pulisher or liene permitting restrited opying in the United Kingdom issued y the opyright Liensing geny Ltd, Sffron House, 6 10 Kiry Street, London E1N 8TS. ll trdemrks used herein re the property of their respetive owners. The use of ny trdemrk in this text does not vest in the uthor or pulisher ny trdemrk ownership rights in suh trdemrks, nor does the use of suh trdemrks imply ny ffilition with or endorsement of this ook y suh owners. ISN 10: ISN 13: ritish Lirry tloguing-in-pulition Dt tlogue reord for this ook is ville from the ritish Lirry Printed in the United Sttes of meri
3 Vetors nd Olique Tringles SE 2: TWO SIDES ND THE NGLE OPPOSITE ONE OF THEM For tringle in whih we know two sides nd the ngle opposite one of the given sides, the solution will e either one tringle, or two tringles, or even possily no tringle. The following exmples illustrte how eh of these results is possile = 40.0 ' Side rehes t either of two points () Fig. 57 Fig () 60.0 ' ' () EXMPLE 4 se 2: Two sides nd ngle opposite Solve the tringle with the following given prts: = 60.0, = 40.0, nd =. First, mke good sle drwing (Fig. 57()) y drwing ngle nd mesuring off 60 for. This will more lerly show tht side = 40.0 will interset side t either position or. This mens there re two tringles tht stisfy the given vlues. Using the lw of sines, we solve the se for whih is n ute ngle: 40.0 ' sin = 40.0 sin = sin -1 or Therefore, = 48.6 nd = Using the lw of sines gin to find, we hve Thus, = 48.6, = 101.4, nd = See Fig. 57(). The other solution is the se in whih, opposite side, is n otuse ngle. Therefore, = = 18.6 Using the lw of sines to find, we hve = = sin = 40.0 sin =180 - = = sin 18.6 = 40.0 sin = 60.0 sin = = 40.0 sin 18.6 sin = sin sin = 78.4 sin = 60.0 sin 40.0 This mens tht the seond solution is =131.4, =18.6, nd =25.5. See Fig. 57(). The omplete sequene for the lultor solution is shown in Fig. 58. The upper window shows the ompletion of the solution for,, nd. The lower window shows the solution for,, nd. n 311
4 Vetors nd Olique Tringles EXMPLE 5 se 2: Possile solutions In Exmple 4, if , only one solution would result. In this se, side would interept side t. It lso interepts the extension of side, ut this would require tht ngle not e inluded in the tringle (see Fig. 59). Thus, only one solution my result if 7. In Exmple 4, there would e no solution if side were not t lest If this were the se, side would not e long enough to even touh side. It n e seen tht must t lest equl sin. If it is just equl to sin, there is one solution, right tringle. See Figure 60. Fig ' Side rehes ut too long for seond Fig Just touhes n miguous se Summrizing the results for se 2 s illustrted in Exmples 4 nd 5, we mke the following onlusions. Given sides nd nd ngle (ssuming here tht nd re orresponding prts), we hve the following summry of solutions for se 2. Prtie Exerise 2. Determine whih of the four possile solution types ours if = 28, = 48, nd = 30. UTION SUMMRY OF SOLUTIONS: TWO SIDES ND THE NGLE OPPOSITE ONE OF THEM 1. No solution if 6 sin. See Fig. 61(). 2. right tringle solution if = sin. See Fig. 61(). 3. Two solutions if sin 6 6. See Fig. 61(). 4. One solution if 7. See Fig. 61(d). ' () () () ' (d) Fig. 61 NOTE NOTE Note tht in order to hve two solutions, we must know two sides nd the ngle opposite one of the sides, nd the shorter side must e opposite the known ngle. If there is no solution, the lultor will indite n error. If the solution is right tringle, the lultor will show n ngle of extly 90 (no extr deiml digits will e displyed). For the reson tht two solutions my result for se 2, it is lled the miguous se. However, it must lso e kept in mind tht there my only e one solution. reful hek of the given prts must e mde in order to determine whether there is one solution or two solutions. The following exmple illustrtes se 2 in n pplied prolem. 312
5 Vetors nd Olique Tringles n See the text introdution. Hvn Est 43.2 v pg 43.2 South Kingston v w 40.0 km/h v p 300 km/h Heding Fig. 62 Prtie Exerise 3. In Exmple 6, wht should e the heding if 300 km> h is hnged to 500 km> h? EXMPLE 6 se 2: pplition Kingston, Jmi, is 43.2 south of est of Hvn, u. Wht should e the heding of plne from Hvn to Kingston if the wind is from the west t 40.0 km>h nd the plne s speed with respet to the ir is 300 km>h? The heding should e set so tht the resultnt of the plne s veloity with respet to the ir v p nd the veloity of the wind v w will e in the diretion from Hvn to Kingston. This mens tht the resultnt veloity v pg of the plne with respet to the ground must e t n ngle of 43.2 south of est from Hvn. Using the given informtion, we drw the vetor tringle shown in Fig. 62. In the tringle, we know tht the ngle t Kingston is 43.2 y noting the lternte-interior ngles. y finding u, the required heding n e found. There n e only one solution, euse v p 7 v w. Using the lw of sines, we hve known side opposite required ngle 40.0 sin u = 300 sin 43.2 sin u = 40.0 sin 43.2, 300 known side opposite known ngle u = 5.2 Therefore, the heding should e = 48.4 south of est. n If we try to use the lw of sines for se 3 or se 4, we find tht we do not hve enough informtion to omplete ny of the rtios. These ses n, however, e solved y the lw of osines s shown in the next setion. EXMPLE 7 ses 3&4 not solvle y lw of sines Given (se 3) two sides nd the inluded ngle of tringle = 2, = 3, = 45, nd (se 4) the three sides (se 4) = 5, = 6, = 7, we set up the rtios (se 3) 2 sin = 3 sin = sin 45, nd (se 4) 5 sin = 6 sin = 7 sin The solution nnot e found euse eh of the three possile equtions in either se 3 or se 4 ontins two unknowns. n EXERISES 5 In Exerises 1 nd 2, solve the resulting tringles if the given hnges re mde in the indited exmples of this setion. 1. In Exmple 2, solve the tringle if the vlue of is hnged to In Exmple 4, solve the tringle if the vlue of is hnged to In Exerises 3 22, solve the tringles with the given prts. 3. = 45.7, = 65.0, = = 3.07, = 26.0, = = 4380, = 37.4, = = 932, = 0.9, = = 4.601, = 3.107, = = 362.2, = 294.6, = = 7751, = 3642, = = 150.4, = 250.9, = = , = 51.0, = = 729, = 121.0, = = 63.8, = 58.4, = = 0.130, = 55.2, = = 4384, = 47.43, = = 283.2, = 13.79, = = 5.240, = 4.446, = = 89.45, = 37.36, = = 2880, = 3650, = = 0.841, = 0.965, = = 450, = 1260, = = 20, = 10, =
6 Vetors nd Olique Tringles In Exerises 23 40, use the lw of sines to solve the given prolems. 23. smll islnd is pproximtely tringle in shpe. If the longest side of the islnd is 520 m, nd two of the ngles re 45 nd 55, wht is the length of the shortest side? 24. ot followed tringulr route going from dok, to dok, to dok, nd k to dok. The ngles turned were 135 t nd 125 t. If is 875 m from, how fr is it from to? 25. The loding rmp t delivery servie is 12.5 ft long nd mkes 18.0 ngle with the horizontl. If it is repled with rmp 22.5 ft long, wht ngle does the new rmp mke with the horizontl? 26. In n eril photo of tringulr field, the longest side is 86.0 m, the shortest side is 52.5 m, nd the lrgest ngle is The sle is 1 m = 2 m. Find the tul length of the third side of the field. 27. The Pentgon (hedqurters of the U.S. Deprtment of Defense) is the lrgest offie uilding in the world. It is regulr pentgon (five sides), 921 ft on side. Find the gretest stright-line distne from one point on the outside of the uilding to nother outside point (the length of digonl). 28. Two ropes hold 175-l rte s shown in Fig. 63. Find the tensions T 1 nd T 2 in the ropes. (Hint: Move vetors so tht they re til to hed to form tringle. The vetor sum T 1 + T 2 must equl 175 l for equilirium.) Fig T l rte 29. Find the tension T in the left guy wire tthed to the top of the tower shown in Fig. 64. (Hint: The horizontl omponents of the tensions must e equl nd opposite for equilirium. Thus, move the tension vetors til to hed to form tringle with vertil resultnt. This resultnt equls the upwrd fore t the top of the tower for equilirium. This lst fore is not shown nd does not hve to e lulted.) T Find the distne etween Grvois ve. nd Jefferson ve. long rsenl St. in St. Louis, from Fig. 66. Fig. 66 Grvois ve mi 50.5 rsenl St. Stellite 0.88 mi Jefferson ve. 32. When n irplne is lnding t n 8250-ft runwy, the ngles of depression to the ends of the runwy re 10.0 nd How fr is the plne from the ner end of the runwy? 33. Find the totl length of the pth of the lser em tht is shown in Fig. 67. Fig m Refletors 34. In widening highwy, it is neessry for onstrution rew to ut into the nk long the highwy. The present ngle of elevtion of the stright slope of the nk is 23.0, nd the new ngle is to e 38.5, leving the top of the slope t its present position. If the slope of the present nk is 220 ft long, how fr horizontlly into the nk t its se must they dig? 35. ommunitions stellite is diretly ove the extension of line etween reeiving towers nd. It is determined from rdio signls tht the ngle of elevtion of the stellite from tower is 89.2, nd the ngle of elevtion from tower is See Fig. 68. If nd re 1290 km prt, how fr is the stellite from? (Neglet the urvture of the erth.) Fig. 64 Fig. 65 Memphis T 330 mi 640 mi 136 tlnt N Find the distne from tlnt to Rleigh, North rolin, from Fig. 65. Rleigh Fig km 36. n stronut on the moon drives lunr rover 16 km in the diretion 60.0 north of est from the se. () Through wht ngle must the rover then e turned so tht y driving 12 km frther the stronut n turn gin to return to se long north-south line? () How long is the lst leg of the trip? () n the stronut mke it k to se if the mximum rnge of the rover is 40 km? 314
7 Vetors nd Olique Tringles 37. ot owner wishes to ross river 2.60 km wide nd go diretly to point on the opposite side 1.75 km downstrem. The ot goes 8.00 km>h in still wter, nd the strem flows t 3.50 km>h. Wht should the ot s heding e? 38. motorist trveling long level highwy t 75 km>h diretly towrd mountin notes tht the ngle of elevtion of the mountin top hnges from out 20 to out 30 in 20-min period. How muh loser on diret line did the mountin top eome? 39. hillside is inlined t 23 with the horizontl. From given point on the slope, it hs een found tht vein of gold is 55 m diretly elow. t wht ngle elow the hillside slope from nother point downhill must stright 65-m shft e dug to reh the vein? 40. Point P on the mehnism shown in Fig. 69 is driven k nd forth horizontlly. If the minimum vlue of ngle u is 32.0, wht is the distne etween extreme positions of P? Wht is the mximum possile vlue of ngle u? Fig. 69 nswers to Prtie Exerises 36.0 m 24.5 m 1. = Two solutions P 6 The Lw of osines Lw of osines se 3: Two Sides & Inluded ngle se 4: Three Sides Summry of Solving Olique Tringles () () h Fig. 70 x x h s noted in the lst setion, the lw of sines nnot e used for se 3 (two sides nd the inluded ngle) nd se 4 (three sides). In this setion, we develop the lw of osines, whih n e used for ses 3 nd 4. fter finding nother prt of the tringle using the lw of osines, we will often find it esier to omplete the solution using the lw of sines. onsider ny olique tringle for exmple, either tringle shown in Fig. 70. For eh tringle, h> = sin, or h = sin. lso, using the Pythgoren theorem, we otin 2 = h 2 + x 2 for eh tringle. Therefore (with (sin ) 2 = sin 2 ), 2 2 sin 2 x 2 (9) In Fig. 70(), note tht 1 - x2> = os, or - x = os. Solving for x, we hve x = - os. In Fig. 70(), + x = os, nd solving for x, we hve x = os -. Sustituting these reltions into Eq. (9), we otin 2 2 sin 2 ( os ) 2 nd 2 2 sin 2 ( os ) 2 respetively. When expnded, these oth give 2 2 sin 2 2 os os 2 (sin 2 os 2 ) 2 2 os Relling the definitions of the trigonometri funtions, we know tht sin u = y>r nd os u = x>r. Thus, sin 2 u + os 2 u = 1y 2 + x 2 2>r 2. However, x 2 + y 2 = r 2, whih mens sin 2 U os 2 U 1 (12) This eqution is vlid for ny ngle u, sine we hve mde no ssumptions s to the properties of u. Thus, y sustituting Eq. (12) into Eq. (11), we rrive t the lw of osines: (10) (11) Lw of osines 2 = os (13) Using the method ove, we my lso show tht 2 = os 2 = os 315
Section 6.1 Law of Sines. Notes. Oblique Triangles - triangles that have no right angles. A c. A is acute. A is obtuse
Setion 6.1 Lw of Sines Notes. Olique Tringles - tringles tht hve no right ngles h is ute h is otuse Lw of Sines - If is tringle with sides,, nd, then sin = sin = sin or sin = sin = sin The miguous se (SS)
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationREVIEW QUESTIONS TOPIC 5 TRIGONOMETRY I FLUENCY
TOPIC 5 TRIGONOMETRY I REVIEW QUESTIONS FLUENCY The most urte mesure for the length of the third side in the tringle elow is: A 4.83 m B 23.3 m C 3.94 m D 2330 mm E 4826 mm 2 Wht is the vlue of x in this
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides - Solving Generl Tringles - Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More information8.1. The Sine Law. Investigate. Tools
8.1 Te Sine Lw Mimi 50 ermud Tringle ermud 1600 km Sn Jun 74 Puerto Rio Te ermud Tringle, in te nort tlnti Oen, is te lotion of severl unexplined plne nd sip disppernes. Vrious teories ve een suggested
More informationPatterns and Algebra
Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing
More informationc The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?
3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem
More informationMacroscopic and Microscopic Springs Procedure
Mrosopi nd Mirosopi Springs Proedure OBJECTIVE Purpose In this l you will: investigte the spring-like properties of stright wire, disover the strethiness of mteril, independent of the size nd shpe of n
More informationNotes on Spherical Triangles
Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The
More informationREVIEW, pages
REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:
More informationResistors, Current and Voltage measurements, Ohm s law, Kirchhoff s first and second law. Kirchhoff s first Objectives:
EE -050 Ciruit L Experiment # esistors, Current nd Voltge mesurements, Ohm s lw, Kirhhoff s first nd seond lw. Kirhhoff s first Ojetives: Slmn in Adul Aziz University Eletril Engineering Deprtment. Fmiliriztion
More informationProbability and Statistics P(A) Mathletics Instant Workbooks. Copyright
Proility nd Sttistis Student Book - Series K- P(A) Mthletis Instnt Workooks Copyright Student Book - Series K Contents Topis Topi - Review of simple proility Topi - Tree digrms Topi - Proility trees Topi
More informationDouble Integrals over Rectangles
Jim Lmbers MAT 8 Spring Semester 9- Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In single-vrible lulus, the definite integrl of
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationSLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises
SLOVAK UNIVERSITY OF TECHNOLOGY Fulty of Mteril Siene nd Tehnology in Trnv ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory exerises Róbert Riedlmjer TRNAVA 00 ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory
More informationDefining the Rational Numbers
MATH10 College Mthemtis - Slide Set 2 1. Rtionl Numers 1. Define the rtionl numers. 2. Redue rtionl numers.. Convert etween mixed numers nd improper frtions. 4. Express rtionl numers s deimls.. Express
More informationFubini for continuous functions over intervals
Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =
More informationLecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationISM-PRO SOFTWARE DIGITAL MICROSCOPE OPERATION MANUAL
MN-ISM-PRO-E www.insize.om ISM-PRO SOFTWARE DIGITAL MICROSCOPE OPERATION MANUAL Desription Clik Next. As the following piture: ISM-PRO softwre is for ISM-PM00SA, ISM-PM600SA, ISM- PM60L digitl mirosopes.
More informationGeometric quantities for polar curves
Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions
More informationTRIGONOMETRIC APPLICATIONS
HPTER TRIGONOMETRI PPLITIONS n ocen is vst expnse tt cn e life-tretening to person wo experiences disster wile oting. In order for elp to rrive on time, it is necessry tt te cost gurd or sip in te re e
More informationSpherical Geometry. This is an article from my home page:
Spheril Geometry This is n rtile from my home pge: www.olewitthnsen.dk Ole Witt-Hnsen nov. 6 Contents. Geometry on sphere.... Spheril tringles...3. Polr tringles...4 3. The right-ngle spheril tringle...6
More informationAGA56... Analog Input Modules. Siemens Building Technologies HVAC Products
7 922 nlog Input odules G56... nlog input modules for the ontrol of SQ5... ir dmper tutors y ontinuous nlog ontrol signls, suh s 4...20 m, nd ontinuous nlog position feedk signls. For supplementry Dt Sheets,
More informationProposed Cable Tables for SAS2
Tle 50 Requirements for internl le ssemlies using SASDrive onnetors n kplnes. Requirement, Units 1,5 Gps 3Gps 6 Gps Bulk le or kplne:, Differentil impene ohm 100 ± 10 100 g Common-moe impene ohm 32,5 ±
More informationMultivariable integration. Multivariable integration. Iterated integration
Multivrible integrtion Multivrible integrtion Integrtion is ment to nswer the question how muh, depending on the problem nd how we set up the integrl we n be finding how muh volume, how muh surfe re, how
More informationSERVICE MANUAL 9940/20/10
9940/0/0 9/0/00 Tle of ontents REMOVING THE EXTERIOR PARTS... A. Light Cover... B. Bse... C. Top Cover (Flp)... 4 D. Hinge Cover... 4 E. Thred Tension Cover... 4 F. Side Enlosure... 5 G. Hndle nd Unit
More informationAnalog Input Modules
7 922 nlog Input odules G56... nlog input modules for the ontrol of SQ5... ir dmper tutors y ontinuous nlog ontrol signls, suh s 4...20 m, nd ontinuous nlog position feedk signls. For supplementry Dt Sheets,
More informationAbdominal Wound Closure Forceps
Inventor: Crlson, Mrk A. My 25, 2007 Adominl Wound Closure Foreps Astrt. The devie is modifition of stndrd tissue foreps for use during losure of dominl wounds mde for surgil proedure. The modifition onsists
More informationSAMPLE. End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte 2 Complete the missing numers in the sequenes elow. 50 25 00 75 8 30 3 28 2 9 Put irle round two of the shpes elow whih hve 3 shded. 3
More information10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More informationSequential Logic (2) Synchronous vs Asynchronous Sequential Circuit. Clock Signal. Synchronous Sequential Circuits. FSM Overview 9/10/12
9//2 Sequentil (2) ENGG5 st Semester, 22 Dr. Hden So Deprtment of Electricl nd Electronic Engineering http://www.eee.hku.hk/~engg5 Snchronous vs Asnchronous Sequentil Circuit This Course snchronous Sequentil
More informationINSTALLATION & OPERATION INSTRUCTIONS LEVER HANDLE LOCKSETS.
INSTALLATION & OPERATION INSTRUCTIONS FOR LEVER HANDLE LOCKSETS 999-00333E_EN FOR BRINKS HOME SECURITY INTERIOR LOCKING & NON-LOCKING LEVER HANDLE LOCKSETS. FITS DOORS 1-3/8" (35 mm) TO 1-3/4" (45 mm)
More information3/8" Square Multi-Turn Cermet Trimmer
Vishy Sfernie 3/8" Squre Multi-Turn Cermet Trimmer FEATURES Industril grde W t 70 C The T93 is smll size trimmer - 3/8" x 3/8" x 3/16" - nswering PC ord mounting requirements. Five versions re ville whih
More information+ sin bsin. sin. tan
6. Spheril rignmetri Frmule Just s in plne gemetry, there re useful trignmetri frmule whih relte the sides nd vertex ngles f spheril tringles: Csine Frmul [6.1] s ss + sin sin s Sine Frmul [6.] sin sin
More information3/8" Square Multi-Turn Cermet Trimmer
www.vishy.om 3/8" Squre Multi-Turn Cermet Trimmer Vishy Sfernie ermet element. FEATURES Industril grde The is smll size trimmer - 3/8" x 3/8" x 3/16" - nswering PC ord mounting requirements. Five versions
More informationSpiral Tilings with C-curves
Spirl Tilings with -curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More information1/4" Multi-Turn Fully Sealed Container Cermet Trimmer
www.vishy.om Vishy Sfernie 1/4" Multi-Turn Fully Seled Continer Cermet Trimmer Due to their squre shpe nd smll size (6.8 mm x 6.8 mm x 5 mm), the multi-turn trimmers of the series re idelly suited for
More informationECE 274 Digital Logic Spring Digital Design. Combinational Logic Design Process and Common Combinational Components Digital Design
ECE 27 Digitl Logi Spring 29 Comintionl Logi Design Proess n Common Comintionl Components Digitl Design 2.7 2. Digitl Design Chpter 2: Comintionl Logi Design Slies to ompn the tetook Digitl Design, irst
More informationASY P.O. BOX 729 TERRELL, TEXAS / PAGE 1 OF 13 SAM
203 Madix Inc., ll rights reserved ommon Parts 2 MXI GRI WIRE GRI SHELF WITH (GPWGS) MXI GRI FIXTURE PNEL (GPWFP) FIXTURE PNELS RE USE S EN SUPPORT. SHELF N E USE NYWHERE. MXI GRI REINFORMENT R 3 (GPR)
More informationSo Many Possibilities page 1 of 2
Otober Solving Problems Ativities & So Mny Possibilities pge of Use the blnk spe to solve eh problem. Show ll your work inluding numbers, words, or lbeled skethes. Write omplete sentene below your work
More informationGLONASS PhaseRange biases in RTK processing
ASS PhseRnge ises in RTK proessing Gle Zyrynov Ashteh Workshop on GSS Bises 202 Bern Switzerlnd Jnury 8-9 202 Sope Simplified oservtion models for Simplified oservtion models for ASS FDMA speifi: lok nd
More informationBalancing Your Life. Ideas that might help you
Blning Your Life Ides tht might help you Pul Hoskin Summer 2007 Let s e honest if one lists off the responsiilities nd hoies tht eh of us hve nd ssigns weekly hourly time tht eh needs to e fulfilled, then
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd
More information1/4" Multi-Turn Fully Sealed Container Cermet Trimmer
1/4" Multi-Turn Fully Seled Continer Cermet Trimmer Due to their squre shpe nd smll size (6.8 mm x 6.8 mm x 5 mm), the multi-turn trimmers of the series re idelly suited for PCB use, enling high density
More informationMETHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin
METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or
More informationCOMPUTER NETWORK DESIGN Network layer protocols
OMPUTER NETWORK ESIGN Network lyer protools Network lyer (lyer 3) Gruppo Reti TL nome.ognome@polito.it http://www.telemti.polito.it/ OMPUTER NETWORK ESIGN Review of network lyer protools - opyright This
More informationDomination and Independence on Square Chessboard
Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion
More informationSolutions to exercise 1 in ETS052 Computer Communication
Solutions to exercise in TS52 Computer Communiction 23 Septemer, 23 If it occupies millisecond = 3 seconds, then second is occupied y 3 = 3 its = kps. kps If it occupies 2 microseconds = 2 6 seconds, then
More informationITEC2620 Introduction to Data Structures
/5/20 ITEC220 Introdution to Dt Strutures Leture 0 Gme Trees Two-Plyer Gmes Rules for gme define the sttespe Nodes re gme sttes Links re possile moves Build serh tree y rute fore Exmple I Exmple II A Our
More information& Y Connected resistors, Light emitting diode.
& Y Connected resistors, Light emitting diode. Experiment # 02 Ojectives: To get some hndson experience with the physicl instruments. To investigte the equivlent resistors, nd Y connected resistors, nd
More informationAcademic. Grade 9 Assessment of Mathematics SAMPLE ASSESSMENT QUESTIONS
Aemi Gre 9 Assessment of Mthemtis 211 SAMPLE ASSESSMENT QUESTIONS Reor your nswers to the multiple-hoie questions on the Stuent Answer Sheet (211, Aemi). Plese note: The formt of this ooklet is ifferent
More informationPolar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationThe Nottingham eprints service makes this work by researchers of the University of Nottingham available open access under the following conditions.
Remenyte-Presott, Rs nd Andrews, John (27) Prime implints for modulrised non-oherent fult trees using inry deision digrms. Interntionl Journl of Reliility nd Sfety, (4). pp. 446-464. ISSN 479-393 Aess
More informationTRANSIENT VOLTAGE DISTRIBUTION IN TRANSFORMER WINDING (EXPERIMENTAL INVESTIGATION)
IJRET: Interntionl Journl of Reserh in Engineering nd Tehnology ISSN: 2319-1163 TRANSIENT VOLTAGE DISTRIBUTION IN TRANSFORMER WINDING (EXPERIMENTAL INVESTIGATION) Knhn Rni 1, R. S. Goryn 2 1 M.teh Student,
More informationYellowknife km Vancouver km NEL
ic tio n Yellowknife Pr e- Pu bl 1566 km 67.3 Vncouver 112 1870 km hpter 3 tio n cute Tringle Trigonometry ic LERNING GOLS You will be ble to develop your sptil sense by Pr e- Pu bl? Using the sine lw
More informationExercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationTranslate and Classify Conic Sections
TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke
More informationDETAIL A SCALE 1 : 85
ISIONS ZONE LTR. ESRIPTION TE PP - INITIL RELESE PER N NORTH.0 0 EST ISOLTE P (0" X.") 0 9 0 X FOOKE # 0 ENTER ISOLTE P (" X 9") WEST ISOLTE P (0" X.") FOOKE # 0. RWING ONTENTS SHEET : OVERLL LYOUT SHEET
More informationINTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 300Q Model the concept of ddition for sums to 10. (QT N 36) Model the concept of sutrction using numers less thn or equl to 10. (QT N 37) Write ddition
More informationAlgebra Practice. Dr. Barbara Sandall, Ed.D., and Travis Olson, M.S.
By Dr. Brr Sndll, Ed.D., Dr. Melfried Olson, Ed.D., nd Trvis Olson, M.S. COPYRIGHT 2006 Mrk Twin Medi, Inc. ISBN 978-1-58037-754-6 Printing No. 404042-EB Mrk Twin Medi, Inc., Pulishers Distriuted y Crson-Dellos
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More informationDigital Design. Sequential Logic Design -- Controllers. Copyright 2007 Frank Vahid
Digitl Design Sequentil Logic Design -- Controllers Slides to ccompny the tetook Digitl Design, First Edition, y, John Wiley nd Sons Pulishers, 27. http://www.ddvhid.com Copyright 27 Instructors of courses
More information7KH4XLQFXQ; Earth/matriX SCIENCE IN ANCIENT ARTWORK. Charles William Johnson
Erth/mtriX SCIENCE IN ANCIENT ARTWORK 7KH4XLQFXQ; Chrles Willim Johnson Erth/mtriX P.O. Box 231126, New Orlens, Louisin, 70183-1126 2001 Copyrighted y Chrles Willim Johnson www.erthmtrix.om www.the-periodi-tle.om
More informationQuestion Paper Wednesday 13 Thursday 14 January 2010
KEY SKILLS INFORMATION AND COMMUNICATION TECHNOLOGY Level 3 ArtComp [KSI31] Question Pper Wenesy 13 Thursy 14 Jnury 2010 Do NOT open this Question Pper until you re tol to y the invigiltor THERE ARE THREE
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationEBU KNOCKOUT COMPETITIONS
EBU KNOCKOUT COMPETITIONS GENERAL REGULATIONS 1 INTRODUCTION Vrious regultions pply to ll English Bridge Union ompetitions tht involve mthes plyed privtely. These ompetitions omprise: The knokout stges
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More informationPearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
Pearson Education Limited Edinburgh Gate Harlow Essex M20 2JE England and ssociated ompanies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 2014 ll
More informationSamantha s Strategies page 1 of 2
Unit 1 Module 2 Session 3 Smnth s Strtegies pge 1 of 2 Smnth hs been working with vriety of multiplition strtegies. 1 Write n expression to desribe eh of the sttements Smnth mde. To solve 18 20, I find
More informationUnilateral and equitransitive tilings by squares of four sizes
Also ville t http://m-journl.eu ISSN 1855-3966 (printe en.), ISSN 1855-3974 (eletroni en.) ARS MATHEMATICA CONTEMPORANEA 10 (2015) 135 167 Unilterl n equitrnsitive tilings y squres of four sizes Csey Mnn
More informationGENERAL NOTES USE OF DESIGN DATA SHEETS:
GENERL DT: TE FOLLOWING DT IS SSUMED: USE OF DT SEETS: ED REVIEWED INTERNL NGLE OF FRITION OF KFILL SOIL, = 30 TOTL UNIT WEIGT OF KFILL SOIL = 20 PF bf. LULTE TE REQUIRED EIGT "" PER TE "STEM EIGT" NOTE
More informationApplications of a New Property of Conics to Architecture: An Alternative Design Project for Rio de Janeiro Metropolitan Cathedral
Jun V. Mrtín Zorrquino Frneso Grnero odrígue José uis Cno Mrtín Applitions of New Property of Conis to Arhiteture: An Alterntive Design Projet for io de Jneiro Metropolitn Cthedrl This pper desries the
More informationEvaluating territories of Go positions with capturing races
Gmes of No Chne 4 MSRI Pulitions Volume 63, 2015 Evluting territories of Go positions with pturing res TEIGO NAKAMURA In nlysing pturing res, or semeis, we hve een fousing on the method to find whih plyer
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More informationDetection of Denial of Service attacks using AGURI
Detetion of Denil of Servie ttks using AGURI Ryo Kizki Keio Univ. kizki@sf.wide.d.jp Kenjiro Cho SonyCSL kj@sl.sony.o.jp Osmu Nkmur Keio Univ. osmu@wide.d.jp Astrt Denil of Servie ttks is divided into
More informationIntroduction 6 Basics 8 Projects 26. Stitching terms 94. About the author 95
Contents Klmh Books 21027 Crossros Cirle Wukesh, Wisonsin 53186 www.klmh.om/books 2010 Lesley Weiss All rights reserve. Exept for rief exerpts for review, this ook my not e reproue in prt or in whole y
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationGENERAL NOTES USE OF DESIGN DATA SHEETS:
GENERL DT: TE FOLLOWING DT IS SSUMED: USE OF DT SEETS: ED INTERNL NGLE OF FRITION OF KFILL SOIL, = 0 TOTL UNIT WEIGT OF KFILL SOIL = 0 PF bf. LULTE TE REQUIRED EIGT "" PER TE "STEM EIGT" NOTE ON TIS SEET.
More informationby Kathy Brown of The Teacher s Pet
y Kthy Brown of The Teher s Pet Designs y Kthy Brown Pieing y Kthy Brown & Lin Ree Mhine quilte y Crol Hilton www.reroosterfris.om Quilte size: pproximtely 40" x 50" Fris from the Sprinkles (Style #4527)
More information(1) Non-linear system
Liner vs. non-liner systems in impednce mesurements I INTRODUCTION Electrochemicl Impednce Spectroscopy (EIS) is n interesting tool devoted to the study of liner systems. However, electrochemicl systems
More informationExperiment 3: The research of Thevenin theorem
Experiment 3: The reserch of Thevenin theorem 1. Purpose ) Vlidte Thevenin theorem; ) Mster the methods to mesure the equivlent prmeters of liner twoterminl ctive. c) Study the conditions of the mximum
More informationA Development of Embedded System for Speed Control of Hydraulic Motor
AISTPME (2011) 4(4): 35-39 A Development of Embedded System for Speed Control of Hydruli Motor Pornjit P. Edutionl Mehtronis Reserh Group Deprtment of Teher Trining in Mehnil Engineering, KMUTN, ngkok,
More informationMATHEMATICS. Student Booklet
GRADE 6 ASSESSMENT OF READING, WRITING AND MATHEMATICS, 2004 2005 Stuent Booklet MATHEMATICS Plese note: The formt of these ooklets is slightly ifferent from tht use for the ssessment. The items themselves
More informationPerformance Comparison between Network Coding in Space and Routing in Space
Performnce omprison etween Network oding in Spce nd Routing in Spce Yunqing Ye, Xin Hung, Ting Wen, Jiqing Hung nd lfred Uwitonze eprtment of lectronics nd Informtion ngineering, Huzhong University of
More informationPatterns and Relationships
Series Techer Ptterns nd Reltionships opyright 009 3P Lerning. All rights reserved. First edition printed 009 in Austrli. A ctlogue record for this ook is ville from 3P Lerning Ltd. ISBN 978-1-91860-3-4
More informationProgramming Guide. Neurostimulators for Chronic Pain. RestoreSensor, RestoreUltra, RestoreAdvanced, and PrimeAdvanced
Progrmming Guide Neurostimultors for Chroni Pin RestoreSensor, RestoreUltr, RestoreAdvned, nd PrimeAdvned For use with SureSn MRI nd erlier non-suresn Neurostimultion Systems. Overview This guide desries
More informationStudent Book SERIES. Fractions. Name
D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A - April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationRWM4400UH High Performance Hand Held Wireless Microphone System
CH 1 CH 2 CH 3 CH 4 UHF QUAD VOLUME MAX VOLUME MAX VOLUME MAX VOLUME RWM 4400UH MIN MIN MIN CHANNEL 1 CHANNEL 2 CHANNEL 3 CHANNEL 4 RWM4400UH High Performne Hnd Held Wireless Mirophone System OWNER S MANUAL
More informationAutomatic Strategy Verification for Hex
utomti Strtegy Verifition for Hex Ryn B. Hywrd, Broderik rneson, nd Philip Henderson Deprtment of Computing Siene, University of lert, Edmonton, Cnd {hywrd,roderi,ph}@s.ulert. strt. We present onise nd/or-tree
More informationSection 10.2 Graphing Polar Equations
Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The
More informationMAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES
MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine E-mil: rt_science@hotmil.com 1 INTRODUCTION During the considertion
More informationComputational Complexity of a Pop-up Book
omputtionl omplexity of Pop-up ook Ryuhei Uehr Shio Termoto strt Origmi is the enturies-ol rt of foling pper, n reently, it is investigte s omputer siene: Given n origmi ith reses, the prolem to etermine
More informationMulti-beam antennas in a broadband wireless access system
Multi-em ntenns in rodnd wireless ccess system Ulrik Engström, Mrtin Johnsson, nders Derneryd nd jörn Johnnisson ntenn Reserch Center Ericsson Reserch Ericsson SE-4 84 Mölndl Sweden E-mil: ulrik.engstrom@ericsson.com,
More informationHomework #1 due Monday at 6pm. White drop box in Student Lounge on the second floor of Cory. Tuesday labs cancelled next week
Announcements Homework #1 due Mondy t 6pm White drop ox in Student Lounge on the second floor of Cory Tuesdy ls cncelled next week Attend your other l slot Books on reserve in Bechtel Hmley, 2 nd nd 3
More informationStudy Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms
Study Guide # 1 MA 26100 - Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,
More informationPolar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a
Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.
More information