Trigonometry and Constructive Geometry
|
|
- Meghan Shelton
- 5 years ago
- Views:
Transcription
1 Trigonometry nd Construtive Geometry Trining prolems for M term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties of these tringles using the lssil method, in ounterlokwise order. 3. Use primes to lel the figure in logil wy. 4. These tringles re ongruent. Lel them using the lssil method. Use primes. Write down reltionships etween ngles nd sides. 1
2 2 Congruene nd similrity 5. These tringles re similr. Lel them using the lssil method. Write down reltionships etween ngles nd etween sides. Wht is the zoom ftor? Is it igger or smller thn 1? 2 Congruene nd similrity 6. Copy these segments using ruler nd ompss. Don t erse your onstrution lines nd rs. Lel your work B () () B () B 7. Use ruler nd ompss to opy the ngle t. Don t erse your rs or onstrution lines. Lel your work. () () () 8. Mke ongruent opy of this tringle y SSS. Use ruler nd ompss. Don t erse your onstrution lines. Lel your work 9. Use ruler nd ompss to mke ongruent opy of this tringle y SS. Lel your work. Explin whih sides nd ngle you hve opied.
3 Congruene nd similrity Use ruler nd ompss to mke ongruent opy of this tringle y S. Lel your work. Explin whih side nd ngles you hve opied. 11. Use ruler nd ompss to onstrut ounterexmple for. Construt two tringles where is true, ut not onguent. Use ruler nd ompss. Lel your tringles nd write down ll the reltionships. Is the zoom ftor igger or smller thn 1? 12. Give ounterexmple for SS, SS. Show tht hving SS true leds to two solutions, one ongruent, the other not ongruent. Use ruler nd ompss. Don t erse your onstrution lines. 13. Prove the prllelogrm re formul re se ˆ height y doing these steps: () Construt prllelogrm y ruler nd ompss. () Cut the prllelogrm into two tringles. () Copy the two tringles into seprte figures using ruler nd ompss. Lel them using lssil lelling nd primes. (d) Use SSS to prove tht the two tringles re ongruent. Explin why eh step is true. (e) Write onlusion. 14. Do the sme prllelogrm proof s in prolem 13 ut using SS. 15. Prove the prlellogrm re formul using S. Follow the steps of prolem Prove the prllelogrm re formul using S. Follow prolem Use to mke smller similr opy of this tringle. Do it with ruler nd ompss. Lel your work. Write down the reltionships etween sides nd ngles. 18. Use to mke lrger similr opy of this tringle. Use ruler nd ompss. Lel your work. Write down reltionships etween sides nd ngles.
4 4 90 tringles 19. Construt 90 perpendiulr lines going through point. Use ruler nd ompss. Don t erse onstrution lines nd rs. () () () 20. Here is right (90 ) tringle. We usully lel the vertex with the right ngle s C nd the longest side s. Lel the tringle nd onstrut the ltitude line h t C using ruler nd ompss. C B 21. Construt 90 tringle using ruler nd ompss. Let C e the 90 vertex. lso onstrut the ltitude line t C. Lel the verties, ngles nd sides of your figure. Don t erse your rs or onstrution lines tringles 22. Fill in the missing ngle. 23. Fill in the missing ngle nd el the sides with proper trigonometri nmes: hypotenuse, djent side, opposite side.
5 90 tringles Here is 90 tringle. Wht is speil out side? Mke list of things. 25. Here is 90 tringle. Wht is speil out side? Mke list of things. 26. Here is 90 tringle. Wht is speil out side? Mke list of things. 27. Find sin, os, tn in terms of sides,,.
6 6 90 tringles 28. Find s, se, ot in terms of sides,,. 29. Consider the following tringles where 1 nd ă () Whih is igger, tn or tn 1? () Whih is igger, ot or ot 1? Explin why!
7 Speil ngles Consider the following tringles where 1 nd ă () Whih is igger, sin or sin 1? () Whih is igger, os or os 1? Explin why! 31. Lel the missing ngle, lel ll sides with proper nmes nd find ll trigonometri rtios sin, os, tn, s, se, ot. in terms of sides,,. 32. Why do we hve six of these trigonometri rtios, sin, os, tn, s, se nd ot? Why re they importnt? Wht is so speil out them? 33. Explin how we were le to lulte the distne to the str 61 Cygni y using trigonometry. 34. Drw 90 tringle, lel it, nd find reltionship etween sin, os nd tn. 35. Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween sin, os nd Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween tn, se nd Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween ot, s nd 1. 4 Speil ngles 38. Chnge these ngles from rdins into degrees. Figure it out y drwing little irles.
8 8 The unit irle digrm () π{2. () π{6. () 7π. (d) 5π{8. (e) 3π{4. (f) 5π{12. (g) 2π{3. (h) 3π{ Chnge these ngles from degrees into rdins. Figure it out y drwing little irles. () 75. () 300. () 285. (d) 120. (e) 225. (f) 720. (g) 195. (h) Sketh π{6 π{3 π{2 tringle. Mke the hypotenuse 1. Lel ll the ngles nd the lengths of the sides. 41. Sketh tringle. Mke the hypotenuse 1. Lel ll the ngles nd the lengths of the sides. 42. Chek tht psin q 2 ` pos q 2 1 for Chek tht pse q 2 ptn q 2 1 for π{ Chek tht ps q 2 pse q 2 1 for The unit irle digrm lwys rememer: the unit irle hs rdius 1, even if tht is not shown expliitly in the digrm. 45. Wht does unit men in the term unit irle? 46. Construt (with ruler nd ompss): xes, unit ile, ngle. 47. Sketh the sine nd osine lines nd lel them Construt with ruler nd ompss: xes, the unit irle, ngle, osine line nd sine line. 49. Wht is the ehvior of sin nd os s goes to 0 nd 90? ngle sin os Ñ 0 Ñ 90
9 The unit irle digrm Prove tht the length u is the sme s sin. u 51. Prove tht length u is os. u 52. Interpret the mening of pos q 2 ` psin q 2 1 using the unit irle digrm. 53. Sketh the tn nd se lines nd lel them Construt using ruler nd ompss: xes, unit irle, ngle, tngent line nd sent line. 55. Wht is the ehvior of tn nd se s goes to 0 nd 90? ngle tn se Ñ 0 Ñ 90
10 10 The unit irle digrm 56. Prove tht length u is tn. Use similr tringles. u 57. Prove tht length u is se. Use similr tringles. u 58. Interpret the mening of pse q 2 ptn q 2 1 using the unit irle digrm. 59. Sketh the osent nd otngent lines nd lel them Use ruler nd ompss to onstrut xes, the unit irle, ngle, osent line nd otngent line. 61. Wht is the ehvior of s nd ot s goes to 0 nd 90? ngle s ot Ñ 0 Ñ 90
11 The unit irle digrm Prove tht u is s. Use similr tringles. u 63. Prove tht u is ot. Use similr tringles. u 64. Interpret the mening of using the unit irle digrm. ps q 2 pot q Orgnize the six trigonometri funtions ording to their reltionship to the unit irle. Inside the irle: Outside the irle: Prtly in nd prtly out:
12 12 Trigonometri identities 66. Fill in this tle of trig funtion ehviors. It s esy if you keep the unit irle digrm in mind, nd use your imgintion. sin Ñ 0 os Ñ? os Ñ? sin Ñ 1 se Ñ 1 s Ñ? s Ñ? se Ñ 8 tn Ñ 0 ot Ñ? ot Ñ? tn Ñ 8 6 Trigonometri identities 67. Wht do we men y trig identity? Give some exmples. 68. pply Pythgors s lw to tringle BC. Do you get nything new? ny new identities or reltionships etween trig funtions? B C 69. pply Pythgors s lw to tringle BC. Do you get nything new? B C
13 Trigonometri identities pply Pythgors s lw to tringle BC. Do you get nything new? C B 71. Given tht the line through the irle is enter line. Wht re the ngles nd 1? Wht re the ngles, 1, 2, 3? The line through the irle is enter line Wht is the reltionship etween ngles nd 1?
14 14 Trigonometri identities Wht re the reltionships etween ngles, 1, 2, 3? Use SSS ongruene to prove tht if tringle hs two equl sides, then it hs two equl ngles. 76. Prove the very eutiful irle-ngle theorem:
15 Trigonometri identities nd now wht hppens when the ngle 1 is on the other side? Prove reltionship etween nd 1 using the sme ides s you used in prolem 76 nd in prolem 75. 1
Trigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
More informationGM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
More informationPythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides
Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationComparing the Pre-image and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity
More informationTopics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule
Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite
More informationLESSON 11: TRIANGLE FORMULAE
. THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.
More informationMath Lesson 4-5 The Law of Cosines
Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationSimilar Right Triangles
Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems
More informationProportions: A ratio is the quotient of two numbers. For example, 2 3
Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit
More informationPYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS
PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides
More informationGeometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.
Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion
More informationIn right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
More informationPROPERTIES OF TRIANGLES
PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.
More informationSimilarity and Congruence
Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles
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 informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
More informationMATHEMATICS AND STATISTICS 1.6
MTHMTIS N STTISTIS 1.6 pply geometri resoning in solving prolems ternlly ssessed 4 redits S 91031 inding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will
More informationIntermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
More informationSection 2.1 Special Right Triangles
Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem
More information9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1
9.1 Dy 1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1 9.1 Dy 2 Wrm Up Use the Pythgoren
More informationBasic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.
si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)
More informationGEOMETRY OF THE CIRCLE TANGENTS & SECANTS
Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the
More information= x x 2 = 25 2
9.1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1 Geometry 9.1 The Pythgoren Theorem 9.1
More informationPlotting Ordered Pairs Using Integers
SAMPLE Plotting Ordered Pirs Using Integers Ple two elsti nds on geoord to form oordinte xes shown on the right to help you solve these prolems.. Wht letter of the lphet does eh set of pirs nme?. (, )
More informationONLINE PAGE PROOFS. Trigonometry Kick off with CAS 12.2 Trigonometry 12.3 Pythagorean triads
12 12.1 Kik off with S 12.2 Trigonometry 12.3 Pythgoren trids Trigonometry 12.4 Three-dimensionl Pythgors theorem 12.5 Trigonometri rtios 12.6 The sine rule 12.7 miguous se of the sine rule 12.8 The osine
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More information2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.
SECTION 8-1 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationMTH 4-16a Trigonometry
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
More informationSect 10.2 Trigonometric Ratios
86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationTrigonometry. Trigonometry. labelling conventions. Evaluation of areas of non-right-angled triangles using the formulas A = 1 ab sin (C )
8 8 Pythgors theorem 8 Pythgoren trids 8 Three-dimensionl Pythgors theorem 8D Trigonometri rtios 8E The sine rule 8F miguous se of the sine rule 8G The osine rule 8H Speil tringles 8I re of tringles res
More informationMCH T 111 Handout Triangle Review Page 1 of 3
Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:
More information6.2 The Pythagorean Theorems
PythgorenTheorems20052006.nb 1 6.2 The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this
More informationTwo Triads of Congruent Circles from Reflections
Forum Geometriorum Volume 8 (2008) 7 12. FRUM GEM SSN 1534-1178 Two Trids of ongruent irles from Refletions Qung Tun ui strt. Given tringle, we onstrut two trids of ongruent irles through the verties,
More informationNaming the sides of a right-angled triangle
6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the
More information3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.
3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 1 MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) TRINGLE = 2-dimentionl she hving 3 sides nd 3 ngles. HRTERISTI OF TRINGLES I) Every tringle is n enclosed she tht hs these
More informationGeometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 3 / 240. Slide 4 / 240. Slide 6 / 240.
Slide 1 / 240 Slide 2 / 240 New Jerse enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freel ville t www.njtl.org nd is intended for the non-ommeril use of students nd tehers.
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationSomething found at a salad bar
Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve
More informationEllipses. The second type of conic is called an ellipse.
Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationm A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationMAT 1275: Introduction to Mathematical Analysis
MAT 75: Intrdutin t Mthemtil Anlysis Dr. A. Rzenlyum Trignmetri Funtins fr Aute Angles Definitin f six trignmetri funtins Cnsider the fllwing girffe prlem: A girffe s shdw is 8 meters. Hw tll is the girffe
More informationSection 13.1 Right Triangles
Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationLesson 2.1 Inductive Reasoning
Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjeture me eriod te n erises 1 9, determine the ngle mesures. 1. p, q 2., 3., 31 82 p 98 q 28 53 17 79 23 50 4. r, s, 5., 6. t t s r 100 85 100 30 4 7 31 7. s 8. m 9. m s 76 35 m
More informationat its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.
Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More information03. Early Greeks & Aristotle
03. Erly Greeks & Aristotle I. Erly Greeks Topis I. Erly Greeks II. The Method of Exhustion III. Aristotle. Anximnder (. 60 B.C.) to peiron - the unlimited, unounded - fundmentl sustne of relity - underlying
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationLesson 2.1 Inductive Reasoning
Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,
More informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus
ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems
More informationA study of Pythagoras Theorem
CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins
More informationExercise sheet 6: Solutions
Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd
More informationPythagorean Theorem and Trigonometry
Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml
More information50 AMC Lectures Problem Book 2 (36) Substitution Method
0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem
More information8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims
More informationIdentifying and Classifying 2-D Shapes
Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationLesson 5.1 Polygon Sum Conjecture
Lesson 5.1 olgon Sum onjeture me eriod te In erises 1 nd 2, find eh lettered ngle mesure. 1.,,, 2.,,, d, e d, e, f d e e d 97 f 26 85 44 3. ne eterior ngle of regulr polgon mesures 10. Wht is the mesure
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationPythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.
Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationPrecalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as
Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle,
More informationStandard Trigonometric Functions
CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent
More informationThe Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationIMPOSSIBLE NAVIGATION
Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd
More informationHOMEWORK FOR CLASS XII ( )
HOMEWORK FOR CLASS XII 8-9 Show tht the reltion R on the set Z of ll integers defined R,, Z,, is, divisile,, is n equivlene reltion on Z Let f: R R e defined if f if Is f one-one nd onto if If f, g : R
More informationART LESSONS & EXERCISES
RT LESSONS & EXERISES Rmón Gllego. www.diujormon.wordpress.com www.diujormon.wordpress.com English - Pge 1 www.diujormon.wordpress.com English - Pge 2 STRTING UP WITH GEOMETRY. DEFINITION OF GEOMETRY Geometry
More information5Trigonometric UNCORRECTED PAGE PROOFS. ratios and their applications
5Trigonometri rtios nd their pplitions 5.1 Kik off with CS 5.2 Trigonometry of right-ngled tringles 5.3 Elevtion, depression nd erings 5.4 The sine rule 5.5 The osine rule 5.6 rs, setors nd segments 5.7
More informationMathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding.
Mthemtis 10 Pge 1 of 5 Properties of s Pthgoren Theorem 2 2 2 used to find the length of sides of right tringle Tpe of s nd Some s Theorems ngles s Slene Isoseles Equilterl ute - ll ngles re less thn 90
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationVECTOR ALGEBRA. Syllabus :
MV VECTOR ALGEBRA Syllus : Vetors nd Slrs, ddition of vetors, omponent of vetor, omponents of vetor in two dimensions nd three dimensionl spe, slr nd vetor produts, slr nd vetor triple produt. Einstein
More informationObjective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15
Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground
More informationProving the Pythagorean Theorem. Proving the Pythagorean Theorem and the Converse of the Pythagorean Theorem
.5 Proving the Pythgoren Theorem Proving the Pythgoren Theorem nd the Converse of the Pythgoren Theorem Lerning Gols In this lesson, you will: Prove the Pythgoren Theorem using similr tringles. Prove the
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More information