SECTION 72 Law of Cosines


 Branden McDowell
 1 years ago
 Views:
Transcription
1 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished nd the ngles,, nd n e mesured. Show tht d SETION 72 Lw of osines Lw of osines Derivtion Solving the SAS se Solving the SSS se If in tringle two sides nd the inluded ngle re given (SAS) or three sides re given (SSS), the lw of sines nnot e used to solve the tringle neither se involves n ngle nd its opposite side (Fig. 1). Both ses n e solved strting with the lw of osines, whih is the sujet mtter for this setion. FIGURE 1 () SAS se () SSS se Lw of osines Derivtion Theorem 1 sttes the lw of osines. Theorem 1 Lw of osines os os os All three equtions sy essentilly the sme thing.
2 72 Lw of osines 517 The lw of osines is used to solve tringles, given: 1. Two sides nd the inluded ngle (SAS) 2. Three sides (SSS) We will estlish the first eqution in Theorem 1. The other two equtions then n e otined from this one simply y releling the figure. We strt y loting tringle in retngulr oordinte system. Figure 2 shows three typil tringles. For n ritrry tringle loted s in Figure 2, the distneetweentwopoints formul is used to otin (h) 2 (k 0) 2 2 (h ) 2 k 2 h 2 2h 2 k 2 Squre oth sides. (1) FIGURE 2 Three representtive tringles. (h, k) (h, k) (h, k) (, 0) h k h () () () k (, 0) k (h, 0) h 0 (, 0) From Figure 2, we note tht 2 h 2 k 2 Sustituting 2 for h 2 k 2 in eqution (1), we otin h (2) But os h h os Thus, y repling h in eqution (2) with os, we reh our ojetive: os [Note: If is ute, then os is positive; if is otuse, then os is negtive.] Solving the SAS se For the SAS se, strt y using the lw of osines to find the side opposite the given ngle. Then use either the lw of osines or the lw of sines to find seond ngle. Beuse of the simpler omputtions, the lw of sines will generlly e used to find the seond ngle.
3 518 7 Additionl Topis in Trigonometry EXPLOREDISUSS 1 After using the lw of osines to find the side opposite the ngle for SAS se, the lw of sines is used to find seond ngle. Figure 2 () shows tht there re two hoies for seond ngle. (A) If the given ngle is otuse, n either of the remining ngles e otuse? Explin. (B) If the given ngle is ute, then one of the remining ngles my or my not e otuse. Explin why hoosing the ngle opposite the shorter side gurntees the seletion of n ute ngle. () Strting with (sin )/ (sin )/, show tht sin sin 1 (D) Explin why eqution (1) gives us the orret ngle only if is ute. (1) The ove disussion leds to the following strtegy for solving the SAS se: Strtegy for Solving the SAS se Step Find Method 1. Side opposite given ngle Lw of osines 2. Seond ngle (Find the ngle Lw of sines opposite the shorter of the two given sides this ngle will lwys e ute.) 3. Third ngle Sutrt the sum of the mesures of the given ngle nd the ngle found in step 2 from 180. EXAMPLE 1 Solving the SAS se Solve the tringle in Figure 3. FIGURE m m
4 72 Lw of osines 519 Solution Solve for Use the lw of osines: os Solve for os (10.3) 2 (6.45) 2 2(10.3)(6.45) os m Solve for Sine side is shorter thn side, must e ute, nd the lw of sines is used to solve for. sin sin sin sin sin sin sin 32.4 sin Solve for sin. Solve for. Sine is ute, the inverse sine funtion gives us diretly Solve for 180 ( ) 180 ( ) Mthed Prolem 1 Solve the tringle with 77.5, 10.4 feet, nd 17.7 feet. Solving the SSS se Strting with three sides of tringle, the prolem is to find the three ngles. Susequent lultions re simplified if we solve for the otuse ngle first, if present. The lw of osines is used for this purpose. A seond ngle, whih must e ute, n e found using either lw, lthough omputtions re usully simpler with the lw of sines. EXPLOREDISUSS 2 (A) Strting with os, show tht os (2) (B) Does eqution (2) give us the orret ngle irrespetive of whether is ute or otuse? Explin. The ove disussion leds to the following strtegy for solving the SSS se.
5 520 7 Additionl Topis in Trigonometry Strtegy for Solving the SSS se Step Find Method 1. Angle opposite longest Lw of osines side this will tke re of n otuse ngle, if present. 2. Either of the remining ngles, Lw of sines whih will e ute (why?) 3. Third ngle Sutrt the sum of the mesures of the ngles found in steps 1 nd 2 from 180. EXAMPLE 2 Solving the SSS se Solve the tringle with 27.3 meters, 17.8 meters, nd 35.2 meters. Solution Three sides of the tringle re given nd we re to find the three ngles. This is the SSS se. Sketh the tringle (Fig. 4) nd use the lw of osines to find the lrgest ngle, then use the lw of sines to find one of the two remining ute ngles. FIGURE m 27.3 m 35.2 m Sine is the lrgest ngle, we solve for it first using the lw of osines. Solve for os os os os 1 (27.3)2 (17.8) 2 (35.2) 2 2(27.3)(17.8) Solve for os. Solve for. Solve for We now solve for either or, using the lw of sines. We hoose. sin sin
6 72 Lw of osines 521 sin sin sin sin sin Solve for sin. Solve for. is ute. Solve for ( ) 180 ( ) 29.8 Mthed Prolem 2 Solve the tringle with 1.25 yrds, 2.05 yrds, nd 1.52 yrds. EXAMPLE 3 Finding the Side of Regulr Polygon If sevensided regulr polygon is insried in irle of rdius 22.8 entimeters, find the length of one side of the polygon. Solution Sketh figure (Fig. 5) nd use the lw of osines: FIGURE 5 Atully, you only need to sketh the tringle: d d (22.8)(22.8) os d 2(22.8)2 2(22.8) 2 os entimeters Mthed Prolem 3 If n 11sided regulr polygon is insried in irle with rdius 4.63 inhes, find the length of one side of the polygon.
7 522 7 Additionl Topis in Trigonometry Answers to Mthed Prolems ft, 33.3, , 95.0, in. EXERISE 72 The leling in the figure elow is the onvention we will follow in this exerise set. Your nswers to some prolems my differ slightly from those in the ook, depending on the order in whih you solve for the sides nd ngles of given tringle. A 1. Referring to the figure ove, if 47.3, 11.7 entimeters, nd 6.04 entimeters, whih of the two ngles, or, n you sy for ertin is ute nd why? 2. Referring to the figure ove, if 93.5, 5.34 inhes, nd 8.77 inhes, whih of the two ngles, or, n you sy for ertin is ute nd why? Solve eh tringle in Prolems , 5.32 yrds, 5.03 yrds , 6.08 entimeters, 5.25 entimeters , 5.73 millimeters, 10.2 millimeters , 8.44 inhes, 20.3 inhes B 7. Referring to the figure t the eginning of the exerise, if 13.5 feet, 20.8 feet, nd 8.09 feet, then if the tringle hs n otuse ngle, whih ngle must it e nd why? 8. Suppose you re told tht tringle hs sides 12.5 entimeters, 25.3 entimeters, nd 10.7 entimeters. Explin why the tringle hs no solution. Solve eh tringle in Prolems 9 12 if the tringle hs solution. Use deiml degrees for ngle mesure meters, 10.2 meters, 9.05 meters miles, 20.7 miles, 12.2 miles kilometers, 5.30 kilometers, 5.52 kilometers meters, 29.4 meters, 33.7 meters Prolems represent vriety of prolems involving oth the lw of sines nd the lw of osines. Solve eh tringle. If prolem does not hve solution, sy so , 88.9, 15.2 entimeters , 102.3, 6.4 millimeters , 13.8 inhes, 12.5 inhes , 16.4 yrds, 28.2 yrds meters, 6.9 meters, 31.3 meters inhes, 32 inhes, 53 inhes , 11.5 inhes, 14.0 inhes , 25.5 meters, 25.5 meters meters, 42.4 meters, 20.4 meters entimeters, 5.23 entimeters, 8.66 entimeters , 7.23 meters, 6.54 meters , 18.1 meters, 22.6 meters , 12.5 inhes, 7.31 inhes , 35.2 inhes, 25.5 inhes 27. Show, using the lw of osines, tht if 90, then (the Pythgoren theorem). 28. Show, using the lw of osines, tht if 2 2 2, then Show tht for ny tringle, Show tht for ny tringle, os os os os os 31. Give solution to Exmple 3 tht does not use the lw of d 360 osines y showing tht 22.8 sin. 2 14
8 72 Lw of osines Show tht the length d of one side of n nsided regulr polygon, insried in irle of rdius r, is given y 180 d 2r sin. n 41. Anlyti Geometry. If point A in the figure hs oordintes (3, 4) nd point B hs oordintes (4, 3), find the rdin mesure of ngle to three deiml ples. y APPLIATIONS 33. Surveying. To find the length AB of smll lke, surveyor mesured ngle AB to e 96, A to e 91 yrds, nd B to e 71 yrds. Wht is the pproximte length of the lke? 0 A B x 42. Anlyti Geometry. If point A in the figure hs oordintes (4, 3) nd point B hs oordintes (5, 1), find the rdin mesure of ngle to three deiml ples. 43. Engineering. Three irles of rdius 2.03, 5.00, nd 8.20 entimeters re tngent to one nother (see figure). Find the three ngles formed y the lines joining their enters (to the nerest 10). A B 34. Surveying. Suppose the figure for this prolem represents the se of lrge rok outropping on frmer s lnd. If surveyor finds AB 110, A 85 meters, nd B 73 meters, wht is the pproximte length (to one deiml ple) of the outropping? 35. Geometry. Two djent sides of prllelogrm meet t n ngle of nd hve lengths of 3 nd 8 feet. Wht is the length of the shorter digonl of the prllelogrm (to 3 signifint digits)? 36. Geometry. Wht is the length of the longer digonl of the prllelogrm in Prolem 35 (to 3 signifint digits)? 37. Nvigtion. Los Angeles nd Ls Vegs re pproximtely 200 miles prt. A pilot 80 miles from Los Angeles finds tht she is 6 20 off ourse reltive to her strt in Los Angeles. How fr is she from Ls Vegs t this time? (ompute the nswer to 3 signifint digits.) 38. Serh nd Resue. At noon, two serh plnes set out from Sn Frniso to find downed plne in the oen. Plne A trvels due west t 400 miles per hour, nd plne B flies northwest t 500 miles per hour. At 2 P.M. plne A spots the survivors of the downed plne nd rdios plne B to ome nd ssist in the resue. How fr is plne B from plne A t this time (to 3 signifint digits)? 39. Geometry. Find the perimeter of pentgon insried in irle of rdius 12.6 meters. 40. Geometry. Find the perimeter of ninesided regulr polygon insried in irle of rdius 7.09 entimeters. 44. Engineering. Three irles of rdius 2.00, 5.00, nd 8.00 inhes re tngent to eh other (see figure). Find the three ngles formed y the lines joining their enters (to the nerest 10). 45. Geometry. A retngulr solid hs sides s indited in the figure. Find AB to the nerest degree. A 4.3 m 8.1 m 2.8 m 46. Geometry. Referring to the figure, find AB to the nerest degree. 47. Spe Siene. For ommunitions etween spe shuttle nd the White Snds trking sttion in southern New Mexio, two stellites re pled in geosttionry orit, 130 prt reltive to the enter of the Erth nd 22,300 B
9 524 7 Additionl Topis in Trigonometry miles ove the surfe of the Erth (see figure). (A stellite in geosttionry orit remins sttionry ove fixed point on the surfe of the Erth.) Rdio signls re sent from the trking sttion y wy of the stellites to the shuttle, nd vie vers. This system llows the trking sttion to e in ontt with the shuttle over most of the Erth s surfe. How fr to the nerest 100 miles is one of the geosttionry stellites from the White Snds trking sttion W? The rdius of the Erth is 3,964 miles. 48. Spe Siene. A stellite S, in irulr orit round the Erth, is sighted y trking sttion T (see figure). The distne TS is determined y rdr to e 1,034 miles, nd the ngle of elevtion ove the horizon is How high is the stellite ove the Erth t the time of the sighting? The rdius of the Erth is 3,964 miles. S T Horizon S W S R Erth SETION 73 Geometri Vetors Geometri Vetors nd Vetor Addition Veloity Vetors Fore Vetors Resolution of Vetors into Vetor omponents Mny physil quntities, suh s length, re, or volume, n e ompletely speified y single rel numer. Other quntities, suh s direted distnes, veloities, nd fores, require for their omplete speifition oth mgnitude nd diretion. The former re often lled slr quntities, nd the ltter re lled vetor quntities. In this setion we limit our disussion to the intuitive ide of geometri vetors in plne. In Setion 74 we introdue lgeri vetors, first step in the generliztion of onept tht hs frrehing onsequenes. Vetors re widely used in mny res of siene nd engineering. Geometri Vetors nd Vetor Addition v O FIGURE 1 Vetor OP, or v. P A line segment to whih diretion hs een ssigned is lled direted line segment. A geometri vetor is direted line segment nd is represented y n rrow (see Fig. 1). A vetor with n initil point O nd terminl point P (the end with the rrowhed) is denoted y OP. Vetors re lso denoted y oldfe letter, suh s v. Sine it is diffiult to write oldfe on pper, we suggest tht you use n rrow over single letter, suh s v, when you wnt the letter to denote vetor. The mgnitude of the vetor OP, denoted y OP, v or v, is the length of the direted line segment. Two vetors hve the sme diretion if they re prllel nd point in the sme diretion. Two vetors hve opposite diretion if they re prllel nd point in opposite diretions. The zero vetor, denoted y 0 or 0, hs mgni
The area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationLesson 18.2: Right Triangle Trigonometry
Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to nonright tringles (hpter
More informationSection 55 Solving Right Triangles*
55 Solving Right Tringles 379 79. Geometry. The re of retngulr nsided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml
More informationThe Pythagorean Theorem Tile Set
The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right
More informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More informationexcenters and excircles
21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling
More informationIntroduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1
3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rellife
More informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationLesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle
: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs
More informationLesson 18.3: Triangle Trigonometry ( ) : OBTUSE ANGLES
Lesson 1.3: Tringle Trigonometry We now extend te teory of rigt tringle trigonometry to nonrigt or olique tringles. Of te six omponents wi form tringle, tree sides nd tree ngles, te possiilities for omintion
More information2.1 ANGLES AND THEIR MEASURE. y I
.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted
More information8. Hyperbolic triangles
8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested
More informationThank you for participating in Teach It First!
Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you
More information8.2 Trigonometric Ratios
8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationState the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127
ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not
More informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More informationPROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions
PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed
More informationPractice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
More information1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.
. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry
More informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
More informationAngles and Triangles
nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002
dius of the Erth  dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl
More informationSOLVING EQUATIONS BY FACTORING
316 (560) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting
More information4.5 The Converse of the
Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationLesson 12.1 Trigonometric Ratios
Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml
More informationHow to Graphically Interpret the Complex Roots of a Quadratic Equation
Universit of Nersk  Linoln DigitlCommons@Universit of Nersk  Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen
More informationMBF 3C Unit 2 Trigonometry Outline
Dy MF 3 Unit 2 Trigonometry Outline Lesson Title Speifi Expettions 1 Review Trigonometry Solving for Sides Review Gr. 10 2 Review Trigonometry Solving for ngles Review Gr. 10 3 Trigonometry in the Rel
More informationThe Pythagorean Theorem
The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully
More information1. Area under a curve region bounded by the given function, vertical lines and the x axis.
Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationChess and Mathematics
Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x
More information11.2 The Law of Sines
894 Applitions of Trigonometry 11. The Lw of Sines Trigonometry literlly mens mesuring tringles nd with Chpter 10 under our belts, we re more thn prepred to do just tht. The min gol of this setion nd the
More informationSOLVING QUADRATIC EQUATIONS BY FACTORING
6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving
More informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More informationRightangled triangles
13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Rightngled tringles Syllus referene Mesurement 4 Rightngled tringles
More informationCHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.
Mthemtis for Austrli Yer 6  Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright
More informationMathematics Reference Sheet. (Pack of 10)
Mthemtis Referene Sheet (Pk of 0 Mthemtis Referene Sheet Use the informtion elow to nswer questions on the Mthemtis test. Cirle π. Are = Cirumferene = h Surfe Are = +h Retngle Trpezoid Are = lw Perimeter
More informationChapter. Contents: A Constructing decimal numbers
Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting
More informationc b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00
Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More informationLATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS. Crystal basis: The structure of the crystal is determined by
LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS Crystl bsis: The struture of the rystl is determined by Crystl Bsis (Point group) Lttie Geometry (Trnsltionl symmetry) Together, the point group
More information1 Fractions from an advanced point of view
1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning
More informationCalculating Principal Strains using a Rectangular Strain Gage Rosette
Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used
More informationEnd 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 Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More informationSAMPLE. Trigonometric ratios and applications
jetives H P T E R 12 Trigonometri rtios nd pplitions To solve prtil prolems using the trigonometri rtios To use the sine rule nd the osine rule to solve prolems To find the re of tringle given two sides
More informationMaximum area of polygon
Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationTrigonometry & Pythagoras Theorem
Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More information9.1 PYTHAGOREAN THEOREM (right triangles)
Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side
More informationSine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.
 Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help
More informationPYTHAGORAS THEOREM 8YEARS. A guide for teachers  Years 8 9. The Improving Mathematics Education in Schools (TIMES) Project
The Improving Mthemtis Edution in Shools (TIMES) Projet PYTHGORS THEOREM guide for tehers  Yers 8 9 MESUREMENT ND GEOMETRY Module 15 June 2011 8YERS 9 Pythgors theorem (Mesurement nd Geometry: Module
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More informationLines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following:
Lines nd s 1 Use ruler nd pencil to drw: 2 prllel lines 2 intersecting lines c 2 perpendiculr lines 2 Complete the following: drw in the digonls on this shpe mrk the interior s on this shpe c mrk equl
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationLesson 2.1 Inductive Reasoning
Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,
More informationLet us recall some facts you have learnt in previous grades under the topic Area.
6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationPythagoras theorem and trigonometry (2)
HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in rightngled tringles. These
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationIf two triangles are perspective from a point, then they are also perspective from a line.
Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident
More informationHeron s Formula for Triangular Area
Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,
More informationEXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE RealWorld Connection EXAMPLE
 Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You
More informationChapter 10 Geometry: Angles, Triangles and Distance
hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding
More informationModule 5. Threephase AC Circuits. Version 2 EE IIT, Kharagpur
Module 5 Threehse A iruits Version EE IIT, Khrgur esson 8 Threehse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (7), the study of iruits, onsisting of the liner elements resistne,
More informationInterior and exterior angles add up to 180. Level 5 exterior angle
22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, eﬁnitions
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationChapter. Fractions. Contents: A Representing fractions
Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter
More information1 GSW IPv4 Addressing
1 For s long s I ve een working with the Internet protools, people hve een sying tht IPv6 will e repling IPv4 in ouple of yers time. While this remins true, it s worth knowing out IPv4 ddresses. Even when
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
More information4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationGeometry of Crystals. Crystal is a solid composed of atoms, ions or molecules that demonstrate long range periodic order in three dimensions
Geometry of Crystls Crystl is solid omposed of toms, ions or moleules tht demonstrte long rnge periodi order in three dimensions The Crystlline Stte Stte of Mtter Fixed Volume Fixed Shpe Order Properties
More informationGeometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm
Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km),
More informationIn order to master the techniques explained here it is vital that you undertake the practice exercises provided.
Tringle formule mtytringleformule0091 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of
More informationVectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m
. Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion
More informationMeasurement and geometry. 0Measuring shapes and time
Mesurement nd geometry 340 0Mesuring shpes nd time NELSON THINK MATHS for the Austrlin Curriulum8 Contents 10.1 Ares of tringles, retngles nd irles 10.2 Ares of other shpes 10.3 Surfe re 10.4 Volumes of
More informationUNIVERSITY AND WORKSTUDY EMPLOYERS WEBSITE USER S GUIDE
UNIVERSITY AND WORKSTUDY EMPLOYERS WEBSITE USER S GUIDE Tble of Contents 1 Home Pge 1 2 Pge 2 3 Your Control Pnel 3 4 Add New Job (ThreeStep Form) 46 5 Mnging Job Postings (Mnge Job Pge) 78 6 Additionl
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationStudent Access to Virtual Desktops from personally owned Windows computers
Student Aess to Virtul Desktops from personlly owned Windows omputers Mdison College is plesed to nnoune the ility for students to ess nd use virtul desktops, vi Mdison College wireless, from personlly
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationNCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:
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: side opposite to A BC sine of A = sin A = hypotenuse
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationGRADE 4. Fractions WORKSHEETS
GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationProving the Pythagorean Theorem
CONCEPT DEVELOPMENT Mthemtis Assessment Projet CLASSROOM CHALLENGES A Formtive Assessment Lesson Proving the Pythgoren Theorem Mthemtis Assessment Resoure Servie University of Nottinghm & UC Berkeley For
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More informationFURTHER TRIGONOMETRY
0 YER The Improving Mthemtics Eduction in Schools (TIMES) Project FURTHER TRIGONOMETRY MESUREMENT ND GEOMETRY Module 24 guide for techers  Yer 0 June 20 Further Trigonometry (Mesurement nd Geometry: Module
More information