8.2 Trigonometric Ratios


 Nickolas Hicks
 2 years ago
 Views:
Transcription
1 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 the ord: You will e le to find the sine, the osine, nd the tngent of n ute ngle. ell Work: Write eh frtion s deiml rounded to the nerest hundredth. 1. 2/ 2. 7/24 Solve eh eqution. 5.8 x x 8.5 ntiiptory Set: y Similrity, right tringle with given ute ngle, suh s, is similr to every other right tringle with tht sme ute ngle mesure. L X Δ ~ ΔLMN ~ ΔXYZ Y Sine the tringles re similr... The rtio of the short leg to the hypotenuse in eh of these tringles will e the sme LN XZ. LM XY The rtio of the long leg to the hypotenuse in eh of these tringles will e the sme MN YZ. LM XY The rtio of the short leg to the long leg in eh of these tringles will e the sme LN XZ. MN YZ These rtios re lled trigonometri rtios. trigonometri rtio is rtio of the lengths of two sides of right tringle. The three si trigonometri rtios re sine, osine, nd tngent, whih re revited s sin, os, nd tn. Trigonometri Rtios Let e right tringle. The sine, the osine, nd the tngent of the ute ngles re defined s follows. M N Z
2 The sine of n ngle is the rtio of the length of the leg opposite the ngle to the length of the hypotenuse. sin sin The osine of n ngle is the rtio of the length of the leg djent to the ngle to the length of the hypotenuse. os djent Leg os djent Leg The tngent of n ngle is the rtio of the length of the leg opposite the ngle to the length of the leg djent to the ngle. tn tn djent Leg djent Leg n often used method of rememering these is the work SOHHTO. Open the ook to pge 541 nd red exmple 1. Exmple: Write eh trigonometri rtio s frtion nd s deiml rounded to the nerest hundredth.. sin J 60/ os J 11/ J tn K 11/ L d. tn J 60 e. sin K f. os K K White ord tivity: dditionl Prtie: Write eh trigonometri rtio s frtion nd s deiml rounded to the nerest hundredth.. sin Y 5/ os Y 12/ tn Y 5/ X 5 d. sin X 12/ e. os X 5/1 0.8 Y 12 Z f. tn X 12/5 2.4
3 The trigonometri rtios of 0, 45, nd 60 n e found using the speil right tringle formuls. Thus they re NEVER written s rounded off deimls Use ny numer you like for. Let = etermine the mesures of the sides of the tringles.,, 2, 2, 6 Sin 45 = os 45 = 1 2 sin 0 = /6 = ½ sin 60 = os 0 = Tn 45 = / = 1 tn 0 = 6 os 60 = /6 = ½ tn 60 = To find the sin, os, nd/or tn of ll other ngles lultor is neessry. efore you n do ny evlution you must mke sure your lultor is in degree mode. For Grphing lultors or lultors whih hve MOE key: push MOE, use the rrow key to move down to Rdin egree, use the rrow key to move right till egree is highlighted, push ENTER or =. For Nongrphing lultors: look for utton leled RG, push repetedly till you see smll in the disply. There re two types of lultors whih n do trigonometri funtions: lger logi, nd nonlger logi. lger Logi Key Strokes Sin os Tn ( egree ) = Nonlger Logi Key Strokes ( egree ) Sin os Tn Open the ook to pge 541 nd red exmple. Exmple: Use your lultor to find eh trigonometri rtio. Round to the nerest hundredth.. sin os tn White ord tivity: Prtie: Use your lultor to find eh trigonometri rtio. Round to the nerest hundredth.. sin os
4 . tn Trigonometri rtios n e used to find missing side on right tringle given n ute ngle nd side. Open the ook to pge 542 nd red exmple 4. Exmple: Find eh length. Round to the nerest hundredth. (o prolem () s n exmple.).. QR. F Q 10.2 ft 15 R 12.9 m P F 9 Use x to represent. sin 6 = QR/ m E x is djent to < QR = 12.9 sin 6 os 9 = 20/F 10.2 is opposite < QR = m F = 20/os 9 Use tn to solve for x. F = m Tn 15 = 10.2/x = 10.2/tn 15 = 8.07 ft. 6 White ord tivity: Prtie: Find eh length. Round to the nerest hundredth... RQ Q. E 17 m 51 R 9.5 in sin 51 = 17/ sin 42 = RQ/9.5 tn 27 = E/1.6 = m RQ = 6.6 in E = 6.9 m 42 P F m E Open the ook to pge 54 nd red exmple 5. Exmple: The Piltushn in Switzerlnd is the world s steepest rilwy. Its steepest setion mkes n ngle of out 25.6 with the horizontl nd rises out 0.9 km. To the nerest hundredth of kilometer, how long is this setion of the rilwy trk? sin 25.6 = 0.9/ = 0.9/sin km = 2.08 km White ord tivity: Prtie: ontrtor is uilding wheel hir rmp for doorwy tht is 1.2 ft ove the ground. The rmp will mke n ngle of 4.8 with the ground. Find the length of the rmp to the nerest hundredth of foot. Sin 4.8 = 1.2/ = 14.4 ft 1.2 ft
5 ssessment: Student pirs will omplete HEK IT OUT pro. 1 5 from this setion. Independent Prtie: Text: pgs pro. 48 even, 911, even, 1820, even, 280, 242 even, 4.
Right Triangle Trigonometry 8.7
304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd
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 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 informationOVERVIEW Prove & Use the Laws of Sines & Cosines G.SRT.10HONORS
OVERVIEW Prove & Use te Lws of Sines & osines G.SRT.10HONORS G.SRT.10 (HONORS ONLY) Prove te Lws of Sines nd osines nd use tem to solve prolems. No interprettion needed  prove te Lw of Sines nd te Lw
More informationEssential Question What are the Law of Sines and the Law of Cosines?
9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry
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 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 informationThe 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 informationRight Triangle Trigonometry
CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use
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 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 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 informationSECTION 72 Law of Cosines
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
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 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 informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
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 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 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 informationRight Triangle Trigonometry for College Algebra
Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 34) II. The Trigonometri
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 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 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 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 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 informationGeometry Notes SIMILAR TRIANGLES
Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving
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 informationPYTHAGORAS THEOREM. Answers. Edexcel GCSE Mathematics (Linear) 1MA0
Edexel GSE Mthemtis (Liner) 1M0 nswers PYTHGORS THEOREM Mterils required for exmintion Ruler grduted in entimetres nd millimetres, protrtor, ompsses, pen, H penil, erser. Tring pper my e used. Items inluded
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 information15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x
Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest
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 informationThis unit will help you to calculate perimeters and areas of circles and sectors, and to find the radius given the circumference or area.
Get strte 1 Cirles This unit will help you to lulte perimeters n res of irles n setors, n to fin the rius given the irumferene or re. AO1 Flueny hek 1 Roun 4.635 to 2 eiml ples (.p.) 2 Roun 5.849 to 1.p.
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 information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
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 information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
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 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 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 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 informationIt may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º
Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion
More informationFractions: Arithmetic Review
Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this
More informationExample
6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte
More informationASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches.
UNFAMILIAR FUNCTIONS (Chpter 19) 547 B ASYMPTOTES An smptote is line whih funtion gets loser n loser to but never quite rehes. In this ourse we onsier smptotes whih re horizontl or vertil. HORIZONTAL ASYMPTOTES
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 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 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 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 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 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 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 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 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 informationTo apply the Law of Cosines. Key Concept Law of Cosines
86 6 Law of osines ontent Standards G.SRT.11 Understand and apply the... Law of osines... lso G.SRT.10 Ojective To apply the Law of osines c a MTHEMTIL PRTIES In the Solve It, you used right triangle
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 informationSTRAND I: Geometry and Trigonometry. UNIT I2 Trigonometric Problems: Text * * Contents. Section. I2.1 Mixed Problems Using Trigonometry
Mthemtics SKE: STRND I UNIT I Trigonometric Prolems: Text STRND I: Geometry nd Trigonometry I Trigonometric Prolems Text ontents Section * * * I. Mixed Prolems Using Trigonometry I. Sine nd osine Rules
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 information5.6 The Law of Cosines
44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt
More informationSimple Electric Circuits
Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through
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 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 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 informationActivity I: Proving the Pythagorean Theorem (Grade Levels: 69)
tivity I: Proving the Pythgoren Theorem (Grde Levels: 69) Stndrds: Stndrd 7: Resoning nd Proof Ojetives: The Pythgoren theorem n e proven using severl different si figures. This tivity introdues student
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 informationMath 2201 Unit 3: Acute Triangle Trigonometry. Ch. 3 Notes
Rea Learning Goals, p. 17 text. Math 01 Unit 3: ute Triangle Trigonometry h. 3 Notes 3.1 Exploring Siengle Relationships in ute Triangles (0.5 lass) Rea Goal p. 130 text. Outomes: 1. Define an aute triangle.
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 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 informationANGLES AND POLYGONS S Mathematics 10. TEACHING MATERIALS from the STEWART RESOURCES CENTRE. Submitted by: Keith Seidler and Romesh Kachroo
NGLES ND POLYGONS Mathematics 10 Submitted by: Keith Seidler and Romesh Kachroo 1 9 9 3 S 105.10 TEHING MTERILS from the STEWRT RESOURES ENTRE INTRODUTION To meet a need for resources for the new Math
More informationThe theorem of. Pythagoras. Opening problem
The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Threedimensionl prolems [4.6] Opening prolem The Louvre
More information11.1 Conic sections (conics)
. Coni setions onis Coni setions re formed the intersetion of plne with right irulr one. The tpe of the urve depends on the ngle t whih the plne intersets the surfe A irle ws studied in lger in se.. We
More informationQuadrilaterals Here are some examples using quadrilaterals
Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4  Exmple 31: igonls of prllelogrm Given
More informationFunctions A B C D E F G H I J K L. Contents:
Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme oordinte or first omponent. Algeri
More informationFinal Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.
MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts
More informationChapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving
Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut
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 information1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE
PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look
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 informationThe Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.
The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion
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 informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationTRIGONOMETRY OF THE RIGHT TRIANGLE
HPTER 8 HPTER TLE OF ONTENTS 81 The Pythagorean Theorem 82 The Tangent Ratio 83 pplications of the Tangent Ratio 84 The Sine and osine Ratios 85 pplications of the Sine and osine Ratios 86 Solving
More informationQuick Guide to Lisp Implementation
isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled Sepressions. The representtion o n Sepression n e roken into two piees, the
More informationWorksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
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 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 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 informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
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 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 informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
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 informationFive Proofs of an Area Characterization of Rectangles
Forum Geometriorum Volume 13 (2013) 17 21. FORUM GEOM ISSN 15341178 Five Proofs of n re Chrteriztion of Retngles Mrtin Josefsson strt. We prove in five ifferent wys neessry n suffiient onition for onvex
More informationChapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals
Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors
More information1 Line Integrals of Scalar Functions
MA 242  Fll 2010 Worksheet VIII 13.2 nd 13.3 1 Line Integrls of Sclr Functions There re (in some sense) four types of line integrls of sclr functions. The line integrls w.r.t. x, y nd z cn be plced under
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 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 informationIntermediate Algebra with Trigonometry. J. Avery 4/99 (last revised 11/03)
Intermediate lgebra with Trigonometry J. very 4/99 (last revised 11/0) TOPIC PGE TRIGONOMETRIC FUNCTIONS OF CUTE NGLES.................. SPECIL TRINGLES............................................ 6 FINDING
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More information