Chapter 5. Triangles and Vectors


 Andrew Spencer Cunningham
 2 years ago
 Views:
Transcription
1 Chpter 5. Tringles nd Vectors 5.3 The Lw of Sines Lerning Objectives Understnd how both forms of the Lw of Sines re obtined. Apply the Lw of Sines when you know two ngles nd nonincluded side nd if you know two ngles nd the included side. Use the Lw of Sines in relworld nd pplied problems. We hve lerned bout the Lw of Cosines, which is generliztion of the Pythgoren Theorem for nonright tringles. We know tht we cn use the Lw of Cosines when: 1. We know two sides of tringle nd the included ngle (SAS) or 2. We know ll three sides of the tringle (SSS) But, wht hppens if the tringle we re working with doesn t fit either of those scenrios? Here we introduce the Lw of Sines. The Lw of Sines is sttement bout the reltionship between the sides nd the ngles in ny tringle. While the Lw of Sines will yield one correct nswer in mny situtions, there re times when it is mbiguous, mening tht it cn produce more thn one nswer. We will explore the mbiguity of the Lw of Sines in the next section. We cn use the Lw of Sines when: 1. We know two ngles nd nonincluded side (AAS) or 2. We know two ngles nd the included side (ASA) Deriving the Lw of Sines ABC contins ltitude CE, which extends from C nd intersects AB. We will refer to the length of ltitude CE s x. We know tht sina= x b nd sinb= x, by the definition of sine. If we crossmultiply both equtions nd substitute, we will hve the Lw of Sines. 331
2 5.3. The Lw of Sines b(sina)=x nd (sinb)=x ց sina b(sina)=(sinb) = sinb b or ւ sina = b sinb Extending these rtios to ngle C nd side c, we rrive t both forms of the Lw of Sines: Form 1 : (sines over sides) Form 2 : (sides over sines) sina = sinb b = sinc c sina = b sinb = c sinc AAS (AngleAngleSide) One cse where we cn to use the Lw of Sines is when we know two of the ngles in tringle nd nonincluded side (AAS). Exmple 1: Using GMN, G=42, N = 73 nd g=12. Find n. Since we know two ngles nd one nonincluded side(g), we cn find the other nonincluded side(n). sin73 = sin42 n 12 nsin42 = 12sin73 n= 12sin73 sin42 n Exmple 2: Continuing on from Exmple 1, find M nd m. 332
3 Chpter 5. Tringles nd Vectors Solution: M is simply = 65. To find side m, you cn now use either the Lw of Sines or Lw of Cosines. Considering tht the Lw of Sines is bit simpler nd new, let s use it. It does not mtter which side nd opposite ngle you use in the rtio with M nd m. Option 1: G nd g Option 2: N nd n sin65 m = sin42 12 msin42 = 12sin65 m= 12sin65 sin42 m sin65 m = sin msin73 = 17.15sin65 m= 17.15sin65 sin73 m Exmple 3: A business group wnts to build golf course on plot of lnd tht ws once frm. The deed to the lnd is old nd informtion bout the lnd is incomplete. If AB is 5382 feet, BC is 3862 feet, AEB is 101, BDC is 74, EAB is 41 nd DCB is 32, wht re the lengths of the sides of ech tringulr piece of lnd? Wht is the totl re of the lnd? Solution: Before we cn figure out the re of the lnd, we need to figure out the length of ech side. In tringle ABE, we know two ngles nd nonincluded side. This is the AAS cse. First, we will find the third ngle in tringle ABE by using the Tringle Sum Theorem. Then, we cn use the Lw of Sines to find both AE nd EB. ABE = 180 (41+101)=38 sin = sin38 sin 101 AE 5382 = sin41 EB AE(sin 101) = 5382(sin 38) EB(sin 101) = 5382(sin 41) AE = 5382(sin38) EB= 5382(sin41) sin 101 sin 101 AE = f eet EB f eet 333
4 5.3. The Lw of Sines Next, we need to find the missing side lengths in tringle DCB. In this tringle, we gin know two ngles nd nonincluded side (AAS), which mens we cn use the Lw of Sines. First, let s find DBC= 180 (74+32)=74. Since both BDC nd DBC mesure 74, tringle DCB is n isosceles tringle. This mens tht since BC is 3862 feet, DC is lso 3862 feet. All we hve left to find now is DB. sin = sin32 DB DB(sin 74) = 3862(sin 32) DB= 3862(sin32) sin74 DB f eet Finlly, we need to clculte the re of ech tringle nd then dd the two res together to get the totl re. From the lst section, we lerned two re formuls, K = 1 2 bcsina nd Heron s Formul. In this cse, since we hve enough informtion to use either formul, we will use K = 1 2 bcsina since it is less computtionlly intense. First, we will find the re of tringle ABE. Tringle ABE: K = 1 2 (3375.5)(5382)sin41 K = 5,959,292.8 ft 2 Tringle DBC: K = 1 2 (3862)(3862)sin32 K = 3,951,884.6 ft 2 The totl re is 5,959, ,951,884.6=9,911,177.4 ft 2. ASA (AngleSideAngle) The second cse where we use the Lw of Sines is when we know two ngles in tringle nd the included side (ASA). For instnce, in T RI: 334
5 Chpter 5. Tringles nd Vectors T, R, nd i re known T, I,nd r re known R, I, nd t re known In this cse, the Lw of Sines llows us to find either of the nonincluded sides. Exmple 4: (Use the picture bove) In T RI, T = 83, R=24, nd i=18.5. Find the mesure of t. Solution: Since we know two ngles nd the included side, we cn find either of the nonincluded sides using the Lw of Sines. Since we lredy know two of the ngles in the tringle, we cn find the third ngle using the fct tht the sum of ll of the ngles in tringle must equl 180. I = 180 (83+24) I = I = 73 Now tht we know I = 73, we cn use the Lw of Sines to find t. sin = sin83 t t(sin73)=18.5(sin83) t = 18.5(sin83) sin73 t 19.2 Notice how we wit until the lst step to input the vlues into the clcultor. This is so our nswer is s ccurte s possible. Exmple 5: In order to void lrge nd dngerous snowstorm on flight from Chicgo to Bufflo, pilot John strts out 27 off of the norml flight pth. After flying 412 miles in this direction, he turns the plne towrd Bufflo. The ngle formed by the first flight course nd the second flight course is 88. For the pilot, two issues re pressing: 1. Wht is the totl distnce of the modified flight pth? 2. How much further did he trvel thn if he hd styed on course? Solution, Prt 1: In order to find the totl distnce of the modified flight pth, we need to know side x. To find side x, we will need to use the Lw of Sines. Since we know two ngles nd the included side, this is n ASA cse. Remember tht in the ASA cse, we need to first find the third ngle in the tringle. 335
6 5.3. The Lw of Sines MissingAngle=180 (27+88)=65 The sum of ngles in tringle is 180 sin = sin27 x Lw of Sines x(sin 65) = 412(sin 27) Cross multiply x= 412(sin27) sin65 x miles Divide by sin 65 The totl distnce of the modified flight pth is =618.4 miles. Solution, Prt 2: To find how much frther John hd to trvel, we need to know the distnce of the originl flight pth, y. We cn use the Lw of Sines gin to find y. sin = sin88 y y(sin 65) = 412(sin 88) y= 412(sin88) sin65 y miles Lw of Sines Cross multiply Divide by sin 65 John hd to trvel =164.1 miles frther. Solving Tringles The Lw of Sines cn be pplied in mny wys. Below re some exmples of the different wys nd situtions to which we my pply the Lw of Sines. In mny wys, the Lw of Sines is much esier to use thn the Lw of Cosines since there is much less computtion involved. Exmple 6: In the figure below, C=22,BC=12,DC=14.3, BDA=65, nd ABD=11. Find AB. Solution: In order to find AB, we need to know one side in ABD. In BCD, we know two sides nd n ngle, which mens we cn use the Lw of Cosines to find BD. In this cse, we will refer to side BD s c. c 2 = (12)(14.3)cos22 c c 5.5 Lw of Cosines 336
7 Chpter 5. Tringles nd Vectors Now tht we know BD 5.5, we cn use the Lw of Sines to find AB. In this cse, we will refer to AB s x. A=180 (11+65)=104 Tringle Sum Theorem sin 104 = sin x x= 5.5sin65 sin 104 x 5.14 Lw of Sines Cross multiply nd divide by sin104 Exmple 7: A group of forest rngers re hiking through Denli Ntionl Prk towrds Mt. McKinley, the tllest mountin in North Americ. From their cmpsite, they cn see Mt. McKinley, nd the ngle of elevtion from their cmpsite to the summit is 21. They know tht the slope of mountin forms 127 ngle with ground nd tht the verticl height of Mt. McKinley is 20,320 feet. How fr wy is their cmpsite from the bse of the mountin? If they cn hike 2.9 miles in n hour, how long will it tke them to get the bse? Solution: As you cn see from the figure bove, we hve two tringles to del with here: right tringle ( MON) nd nonright tringle ( MOU). In order to find the distnce from the cmpsite to the bse of the mountin, y, we first need to find one side of our nonright tringle, MOU. If we look t M in MNO, we cn see tht side ON is our opposite side nd side x is our hypotenuse. Remember tht the sine function is opposite/hypotenuse. Therefore we cn find side x using the sine function. sin21 = x xsin21 = x= sin21 x Now tht we know side x, we know two ngles nd the nonincluded side in MOU. We cn use the Lw of Sines to solve for side y. First, MOU = = 32 by the Tringle Sum Theorem. sin = sin32 y ysin127 = sin32 y= sin32 sin127 x or 7.1 miles If they cn hike 2.9 miles per hour, then they will hike the 7.1 miles in 2.45 hours, or 2 hours nd 27 minutes. 337
8 5.3. The Lw of Sines Points to Consider Are there ny situtions where we might not be ble to use the Lw of Sines or the Lw of Cosines? Considering wht you lredy know bout the sine function, is it possible for two ngles to hve the sme sine? How might this ffect using the Lw of Sines to solve for n ngle? By using both the Lw of Sines nd the Lw of Cosines, it is possible to solve ny tringle we re given? Review Questions 1. In the tble below, you re given figure nd informtion known bout tht figure. Decide if ech sitution represents the AAS cse or the ASA cse. TABLE 5.4: Given Figure Cse. b=16,a=11.7,c=23.8 b. e=214.9,d=39.7,e = 41.3 c. G=22,I = 18,H = 140 d. k=6.3,j = 16.2,L=40.3 e. M = 31,O=9,m=15 f. Q=127,R=21.8,r= Even though ASA nd AAS tringles represent two different cses of the Lw of Sines, wht do they both hve in common?
9 Chpter 5. Tringles nd Vectors 3. Using the figures nd the given informtion from the tble bove, find the following if possible:. side b. side d c. side i d. side l e. side o f. side q 4. In GHI, I = 21.3, H = 62.1, nd i=108. Find g nd h. 5. Use the Lw of Sines to show tht b = sina sinb is true. 6. Use the Lw of Sines, the Lw of Cosines, nd trigonometry functions to solve for x.. b. 7. In order to void storm, pilot strts out 11 off pth. After he hs flown 218 miles, he turns the plne towrd his destintion. The ngle formed between his first pth nd his second pth is 105. If the plne trveled t n verge speed of 495 miles per hour, how much longer did the modified flight tke? 8. A delivery truck driver hs three stops to mke before she must return to the wrehouse to pick up more pckges. The wrehouse, Stop A, nd Stop B re ll on First Street. Stop A is on the corner of First Street nd Route 52, which intersect t 41 ngle. Stop B is on the corner of First Street nd Min Street, which intersect t 103 ngle. Stop C is t the intersection of Min Street nd Route 52. The driver knows tht Stop A nd Stop B re 12.3 miles prt nd tht the wrehouse is 1.1 miles from Stop A. If she must be bck to the wrehouse by 10:00.m., trvels t speed of 45 MPH, nd tkes 2 minutes to deliver ech pckge, t wht time must she leve? Review Answers ASA 2. AAS 3. neither 4. ASA 5. AAS 6. AAS 2. Student nswers will vry but they should notice tht in both cses you know or cn find n ngle nd the side cross from it. 339
10 5.3. The Lw of Sines sin = sin ,=5.6 sin = sin39.7 d,d = not enough informtion sin l = sin ,l = 4.9 sin9 5. o = sin31 15,o=4.6 sin q = sin ,q= G= = 96.6 sin96.6 g = sin21.3,g= sin62.1 h = sin21.3,h= sina = sinb b (sin B) = b(sin A) b = sina sinb Lw of Sines Cross multiply Divide by b(sinb) 1. tn54 = h 7.15 h=9.8,cos67 = 9.8 x x= The ngle we re finding is the one t the fr left side of the tringle = cosA A=43.4, sin43.4 x = sin x= First we need to find the other two sides in the tringle. sin = sin11 x = sin105 y,x=46.3,y=234.3, where y is the length of the originl fight pln. The modified flight pln is = Dividing both by 495 mi/hr, we get 32 min (modified) nd 28.4 min (originl). Therefore, the modified flight pln is 3.6 minutes longer. 6. First, we need to find the distnce between Stop B (B) nd Stop C (C). sin = sin41 B = sin103 C B=13.7,C = The totl length of her route is =48.6 miles. Dividing this by 45 mi/hr, we get tht it will tke her 1.08 hours, or 64.8 minutes, of ctul driving time. In ddition to the driving time, it will tke her 6 minutes (three stops t 2 minutes per stop) to deliver the three pckges, for totl roundtrip time of 70.8 minutes. Subtrcting this 70.8 minutes from 10:00 m, she will need to leve by 8:49 m. 340
The Law of Sines. Say Thanks to the Authors Click (No sign in required)
The Law of Sines Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
More informationContents TRIGONOMETRIC METHODS PROBABILITY DISTRIBUTIONS
ontents UNIT 7 TRIGONOMETRI METHODS Lesson 1 Trigonometric Functions................... 462 1 onnecting ngle Mesures nd Liner Mesures.............. 463 2 Mesuring Without Mesuring.........................
More informationChp. 3_4 Trigonometry.notebook. October 01, Warm Up. Pythagorean Triples. Verifying a Pythagorean Triple... Pythagorean Theorem
Chp. 3_4 Trigonometry.noteook Wrm Up Determine the mesure of the vrile in ech of the following digrms: x + 2 x x 5 x + 3 Pythgoren Theorem  is fundmentl reltionship mongst the sides on RIGHT tringle.
More informationIn any rightangle triangle the side opposite to the right angle is called the Label the Hypotenuse in each diagram above.
9 Ademi Mth Dte: Pythgoren Theorem RIGHT ANGLE TRIANGLE  A right tringle is tringle with one 90 0 ngle. For exmple: In ny rightngle tringle the side opposite to the right ngle is lled the Lbel the Hypotenuse
More informationINVESTIGATION 2. What s the Angle?
INVESTIGATION 2 Wht s the Angle? In the previous investigtion, you lerned tht when the rigidity property of tringles is comined with the ility to djust the length of side, the opportunities for useful
More informationStarter. The Cosine Rule. What the Cosine Rule is and how to apply it to triangles. I can write down the Cosine Rule from memory.
Strter 1) Find the re of the green tringle. 12.8m 2) 2 4 ( + ) x 3 5 3 2 54.8 o 9.7m The Cosine Rule Tody we re lerning... Wht the Cosine Rule is nd how to pply it to tringles. I will know if I hve een
More informationWhy? DF = 1_ EF = _ AC
Similr Tringles Then You solved proportions. (Lesson 2) Now 1Determine whether two tringles re similr. 2Find the unknown mesures of sides of two similr tringles. Why? Simon needs to mesure the height
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Interior nd Exterior Angles of Tringle Tringle Sum, Exterior Angle, nd Exterior Angle Inequlity Theorems Vocbulry Write the term tht best completes ech
More informationLesson 12.1 Right Triangle Trigonometry
Lesson 12.1 Right Tringle Trigonometr 1. For ech of the following right tringles, find the vlues of sin, cos, tn, sin, cos, nd tn. (Write our nswers s frctions in lowest terms.) 2 15 9 10 2 12 2. Drw right
More informationMTH 112: Elementary Functions
1/14 MTH 112: Elementry Functions Section 8.1: Lw of Sines Lern out olique tringles. Derive the Lw os Sines. Solve tringles. Solve the miguous cse. 8.1:Lw of Sines. 2/14 Solving olique tringles Solving
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Interior nd Exterior Angles of Tringle Tringle Sum, Exterior Angle, nd Exterior Angle Inequlity Theorems Vocbulry Write the term tht best completes ech
More informationSpecial Right Triangles
Pge of 5 L E S S O N 9.6 Specil Right Tringles B E F O R E Now W H Y? Review Vocbulr hpotenuse, p. 465 leg, p. 465 You found side lengths of right tringles. You ll use specil right tringles to solve problems.
More informationLesson 2 PRACTICE PROBLEMS Using Trigonometry in Any Triangle
Nme: Unit 6 Trigonometri Methods Lesson 2 PRTIE PROLEMS Using Trigonometry in ny Tringle I n utilize the Lw of Sines nd the Lw of osines to solve prolems involving indiret mesurement in nonright tringles.
More information8.7 Extension: Laws of Sines and Cosines
www.ck12.org Chapter 8. Right Triangle Trigonometry 8.7 Extension: Laws of Sines and Cosines Learning Objectives Identify and use the Law of Sines and Cosines. In this chapter, we have only applied the
More informationMath commonly used in the US Army Pathfinder School
Mth commonly used in the US Army Pthfinder School Pythgoren Theorem is used for solving tringles when two sides re known. In the Pthfinder Course it is used to determine the rdius of circulr drop zones
More informationRecall that the area of a triangle can be found using the sine of one of the angles.
Nme lss Dte 14.1 Lw of Sines Essentil Question: How n you use trigonometri rtios to find side lengts nd ngle mesures of nonrigt tringles? Resoure Loker Explore Use n re Formul to Derive te Lw of Sines
More informationThe Pythagorean Theorem and Its Converse Is That Right?
The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge, Mrking the Text, Shred Reding, Summrize/Prphrse/Retell ACTIVITY 3.6 How did Pythgors get theorem
More informationApply the Pythagorean Theorem
8. Apply the Pythgoren Theorem The Pythgoren theorem is nmed fter the Greek philosopher nd mthemtiin Pythgors (580500 B.C.E.). Although nient texts indite tht different iviliztions understood this property
More information1 Measurement. What you will learn. World s largest cylindrical aquarium. Australian Curriculum Measurement and Geometry Using units of measurement
Austrlin Curriulum Mesurement nd Geometry Using units of mesurement hpter 1 Mesurement Wht you will lern 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Conversion of units Perimeter Cirumferene Are Are of irle Surfe
More informationRight Triangle Trigonometry
Right Tringle Trigonometry To the ncient Greeks, trigonometry ws the study of right tringles. Trigonometric functions (sine, cosine, tngent, cotngent, secnt, nd cosecnt) cn be defined s right tringle rtios
More information17.3 Find Unknown Side Lengths
? Nme 17.3 Find Unknown Side Lenths ALGEBRA Essentil Question How cn you find the unknown lenth of side in polyon when you know its perimeter? Geometry nd Mesurement 3.7.B MATHEMATICAL PROCESSES 3.1.A,
More informationRight Triangle Trigonometry
ONDENSED 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 informationApply the Law of Sines. You solved right triangles. You will solve triangles that have no right angle.
13.5 pply te Lw of Sines TEKS.1,.4, 2.4.; P.3.E efore Now You solved rigt tringles. You will solve tringles tt ve no rigt ngle. Wy? So you n find te distne etween frwy ojets, s in Ex. 44. Key Voulry lw
More informationSUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Bring to school the 1 st day of class!
SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Bring to school the st d of clss! This summer ssignment is designed to prepre ou for Functions/Trigonometr. Nothing on the summer ssignment is new. Everthing
More informationMATHEMATICAL PRACTICES In the Solve It, you used what you know about triangles to find missing lengths. Key Concept Law of Sines
85 205 Lw of Sines ontent Stndrds G.SRT.11 Understnd nd ppl the Lw of Sines... to find unknown mesurements in right nd nonright tringles... lso G.SRT.10 Ojetives To ppl the Lw of Sines 66 ft 35 135
More informationSUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Due September 7 th
SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Due Septemer 7 th This summer ssignment is designed to prepre ou for Functions/Trigonometr. Nothing on the summer ssignment is new. Everthing is review of topics
More informationERRATA for Guide for the Development of Bicycle Facilities, 4th Edition (GBF4)
Dvid Bernhrdt, P.E., President Commissioner, Mine Deprtment of Trnsporttion Bud Wright, Executive Director 444 North Cpitol Street NW, Suite 249, Wshington, DC 20001 (202) 6245800 Fx: (202) 6245806 www.trnsporttion.org
More informationBicycle wheel and swivel chair
Aim: To show conservtion of ngulr momentum. To clrify the vector chrcteristics of ngulr momentum. (In this demonstrtion especilly the direction of ngulr momentum is importnt.) Subjects: Digrm: 1Q40 (Conservtion
More informationChapter 31 Pythagoras theorem and trigonometry (2)
HPTR 31 86 3 The lengths of the two shortest sides of rightngled tringle re m nd ( 3) m respetively. The length of the hypotenuse is 15 m. Show tht 2 3 108 Solve the eqution 2 3 108 Write down the lengths
More informationThe statements of the Law of Cosines
MSLC Workshop Series: Math 1149 and 1150 Law of Sines & Law of Cosines Workshop There are four tools that you have at your disposal for finding the length of each side and the measure of each angle of
More informationGeometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 4 / 240. Slide 3 / 240. Slide 6 / 240.
Slide 1 / 240 New Jersey enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freely ville t www.njtl.org nd is intended for the nonommeril use of students nd tehers. These mterils
More informationGrade 6. Mathematics. Student Booklet SPRING 2011 RELEASED ASSESSMENT QUESTIONS. Record your answers on the MultipleChoice Answer Sheet.
Grde 6 Assessment of Reding, Writing nd Mthemtics, Junior Division Student Booklet Mthemtics SPRING 211 RELEASED ASSESSMENT QUESTIONS Record your nswers on the MultipleChoice Answer Sheet. Plese note:
More information6 TRIGONOMETRY TASK 6.1 TASK 6.2. hypotenuse. opposite. adjacent. opposite. hypotenuse 34. adjacent. opposite. a f
1 6 TIGONOMETY TK 6.1 In eh tringle elow, note the ngle given nd stte whether the identified side is in the orret position or not. 1. 4. opposite 41 2. djent 3. 58 63 djent 32 hypotenuse 5. 68 djent 6.
More information1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely.
9.7 Warmup 1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely. 2. A right triangle has a leg length of 7 in. and a hypotenuse length of 14 in. Solve the triangle
More information8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle
Chapter 8 Applications of Trigonometry 81 8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle A triangle that is not a right triangle is
More informationGeometry Proofs: Chapter 7, Sections 7.1/7.2
Pythgoren Theorem: Proof y Rerrngement of re Given: Right tringle with leg lengths nd, nd hypotenuse length. Prove: 2 2 2 = + Proof #1: We re given figures I nd II s ongruent right tringles III with leg
More informationChapter 4 Group of Volunteers
CHAPTER 4 SAFETY CLEARANCE, FREEBOARD AND DRAUGHT MARKS 41 GENERAL 41.1 This chpter specifies the minimum freebord for inlnd wterwy vessels. It lso contins requirements concerning the indiction of the
More informationUnit 3 Trigonometry. 3.1 Use Trigonometry to Find Lengths
Topic : Goal : Unit 3 Trigonometry trigonometry I can use the primary trig ratios to find the lengths of sides in a right triangle 3.1 Use Trigonometry to Find Lengths In any right triangle, we name the
More informationPhysics 20 Lesson 12 Relative Motion
Physics 20 Lesson 12 Reltie Motion In Lessons 10 nd 11, we lerned how to dd rious dislcement ectors to one nother nd we lerned numer of methods nd techniques for ccomlishin these ector dditions. Now we
More informationPhysics 20 Lesson 11 Relative Motion
Physics 20 Lesson 11 Reltie Motion In Lessons 9 nd 10, we lerned how to dd rious dislcement ectors to one nother nd we lerned numer of methods nd techniques for ccomlishin these ector dditions. Now we
More information7.2 Assess Your Understanding
538 HPTER 7 pplitions of Trigonometri Funtions 7. ssess Your Understnding re You Prepred? nswers re given t the end of these exerises. If you get wrong nswer, red the pges listed in red. 1. The differene
More informationIn previous examples of trigonometry we were limited to right triangles. Now let's see how trig works in oblique (not right) triangles.
The law of sines. In previous examples of trigonometry we were limited to right triangles. Now let's see how trig works in oblique (not right) triangles. You may recall from Plane Geometry that if you
More informationWorking Paper: Reversal Patterns
Remember to welcome ll ides in trding. AND remember to reserve your opinion until you hve independently vlidted the ide! Working Pper: Reversl Ptterns Working Pper In this pper I wnt to review nd (hopefully)
More informationMath Section 4.1 Special Triangles
Math 1330  Section 4.1 Special Triangles In this section, we ll work with some special triangles before moving on to defining the six trigonometric functions. Two special triangles are 30 60 90 triangles
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 The Pythagorean Theorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number 9 Square Positive Square Root 1 4 1 16 Vocabulary Builder leg (noun)
More information2014 Victorian Shooting Championship
2014 Victorin Shooting Chmpionship VPCI, in conjunction with the Stte Coches nd the Stte Umpires invite ll PFA licensed petnque plyers in the Stte of Victori to tke prt in the 2014 Victorin Shooting Chmpionship.
More information5.5 The Law of Sines
434 HPTER 5 nlyti Trigonometry 5.5 Te Lw of Sines Wt you ll lern out Deriving te Lw of Sines Solving Tringles (S, S) Te miguous se (SS) pplitions... nd wy Te Lw of Sines is powerful extension of te tringle
More informationModule 13 Trigonometry (Today you need your notes)
Module 13 Trigonometry (Today you need your notes) Question to ponder: If you are flying a kite, you know the length of the string, and you know the angle that the string is making with the ground, can
More informationSt Ac Ex Sp TOPICS (Text and Practice Books) 4.1 Triangles and Squares Pythagoras' Theorem  
MEP: Demonstrtion Projet UNIT 4 Trigonometry N: Shpe, Spe nd Mesures e,f St Ex Sp TOPIS (Text nd Prtie ooks) 4.1 Tringles nd Squres    4. Pythgors' Theorem   4.3 Extending Pythgors' Theorem   4.4
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in rightangled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in rightangled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (RightAngled Triangles)
More information8.1 Right Triangle Trigonometry; Applications
SECTION 8.1 Right Tringle Trigonometry; pplitions 505 8.1 Right Tringle Trigonometry; pplitions PREPRING FOR THIS SECTION efore getting strted, review the following: Pythgoren Theorem (ppendix, Setion.,
More informationPRESSURE LOSSES DUE TO THE LEAKAGE IN THE AIR DUCTS  A SAFETY PROBLEM FOR TUNNEL USERS?
 7  PRESSURE LOSSES DUE TO THE LEAKAGE IN THE AIR DUCTS  A SAFETY PROBLEM FOR TUNNEL USERS? Pucher Krl, Grz Uniersity of Technology, Austri EMil: pucherk.drtech@gmx.t Pucher Robert, Uniersity of Applied
More informationPythagorean Theorem Name:
Name: 1. A wire reaches from the top of a 13meter telephone pole to a point on the ground 9 meters from the base of the pole. What is the length of the wire to the nearest tenth of a meter? A. 15.6 C.
More informationToday we will focus on solving for the sides and angles of nonright triangles when given two angles and a side.
5.5 The Law of Sines: Part 1 PreCalculus Learning Targets: 1. Use the Law of Sines to solve nonright triangles. Today we will focus on solving for the sides and angles of nonright triangles when given
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 he Pythagorean heorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root 9 81 3 1 4 1 16 1 2 Vocabulary Builder leg
More informationName Class Date SAMPLE. Complete the missing numbers in the sequences below. 753, ,982. The area of the shape is approximately cm 2
End of term: TEST A You will need penil. Yer 5 Nme Clss Dte 1 2 Complete the missing numers in the sequenes elow. 200 3926 4926 400 500 700 7926 753,982 553,982 Estimte the re of the shpe elow. The re
More information1985 BFS CLINICS. BFS ClinicAssembly in Kamloops, British Columbia. Westsyde High School. Bob Bridges is the Football Coach.
1985 BFS CLINICS BFS ClinicAssembly in Kmloops, British Columbi. Westsyde High School. Bob Bridges is the Footbll Coch. BFS Clinicin Rick Anderson detils BFS Nutrition Progrm t Lke Hvsu High School in
More informationOpen Access Regression Analysisbased Chinese Olympic Games Competitive Sports Strength Evaluation Model Research
Send Orders for Reprints to reprints@benthmscience.e The Open Cybernetics & Systemics Journl, 05, 9, 79735 79 Open Access Regression Anlysisbsed Chinese Olympic Gmes Competitive Sports Strength Evlution
More informationChapter 8: Right Triangles (page 284)
hapter 8: Right Triangles (page 284) 81: Similarity in Right Triangles (page 285) If a, b, and x are positive numbers and a : x = x : b, then x is the between a and b. Notice that x is both in the proportion.
More informationSpecial Right Triangles
GEOMETRY Special Right Triangles OBJECTIVE #: G.SRT.C.8 OBJECTIVE Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. *(Modeling Standard) BIG IDEA (Why is
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in rightangled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in rightangled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (RightAngled Triangles)
More informationApplication of Geometric Mean
Section 81: Geometric Means SOL: None Objective: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse
More informationParking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty?
Parking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty? a plane burger Agenda 1 23 hw? Finish Special Right Triangles L8 3 Trig Ratios HW:
More informationRight Triangles and Trigonometry. Right Triangles and Trigonometry
Right Tringles nd Trigonometr hpter Overview nd Pcing PING (ds) Regulr lock sic/ sic/ verge dvnced verge dvnced Geometric Men (pp. ) 0. 0. Find the geometric men etween two numers. Solve prolems involving
More informationUnit 2 Day 4 Notes Law of Sines
AFM Unit 2 Day 4 Notes Law of Sines Name Date Introduction: When you see the triangle below on the left and someone asks you to find the value of x, you immediately know how to proceed. You call upon your
More informationOVERVIEW Similarity Leads to Trigonometry G.SRT.6
OVERVIEW Similarity Leads to Trigonometry G.SRT.6 G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric
More information9.3 AltitudeonHypotenuse Theorems
9.3 AltitudeonHypotenuse Theorems Objectives: 1. To find the geometric mean of two numbers. 2. To find missing lengths of similar right triangles that result when an altitude is drawn to the hypotenuse
More informationBASICS OF TRIGONOMETRY
Mathematics Revision Guides Basics of Trigonometry Page 1 of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF TRIGONOMETRY Version: 1. Date: 0910015 Mathematics Revision
More informationBASKETBALL SPEED AND AGILITY
SKETLL SPEED ND GILITY Off court Speed and gility Work: ox gility Drills: cone set up 5 yards apart, read and follow description Drill 1 : (1234) Sprint around cones, make hard cuts Drill 2: 12 Sprint,
More informationbark bark bat bat Multiple Meaning Words: Kindergarten to Grade 2 More Teaching Tools at harsh sound made by a dog
the brk the brk bt bt hrsh sound mde by dog Mx, stop brking! outside cover of the trunks, brnches, nd roots of woody plnts The brk of this tree is very rough. club of wood or metl used to hit the bll in
More informationLearning Goal: I can explain when to use the Sine, Cosine and Tangent ratios and use the functions to determine the missing side or angle.
MFM2P Trigonometry Checklist 1 Goals for this unit: I can solve problems involving right triangles using the primary trig ratios and the Pythagorean Theorem. U1L4 The Pythagorean Theorem Learning Goal:
More informationThe Discussion of this exercise covers the following points: The openloop ZieglerNichols method. The openloop ZieglerNichols method
Exercise 63 Level Process Control EXERCISE OBJECTIVE In this exercise, you will perform PID control of level process. You will use the openloop step response method to tune the controller. DISCUSSION
More information1970 BRITISH COHORT STUDY: SURVEY
1970 BRITISH COHORT STUDY: 201618 SURVEY Selfcompletion Questionnire HOW TO FILL IN THE QUESTIONNAIRE Plese complete the questionnire using blck or blue ink. The questionnire will be red by scnner, so
More informationTeeJay Publishers Homework for Level C book Ch 12  Length & Area
Chpter 12 Exerise Perentges 1 Length & Are 1. Would you use ruler, tpe mesure or r odometer to mesure : your tehers height the length of 5 note the length of your edroom d the distne from Glsgow to Crlisle?
More informationLesson 21: Special Relationships within Right Triangles Dividing into Two Similar SubTriangles
: Special Relationships within Right Triangles Dividing into Two Similar SubTriangles Learning Targets I can state that the altitude of a right triangle from the vertex of the right angle to the hypotenuse
More informationLet s go Fly a Kite Up, in the Atmosphere!!!
Let s go Fly a Kite Up, in the Atmosphere!!! For this major grade project, you will be designing, constructing, and flying a kite. You may work in teams of no more than 2 students, from the same class
More information2014 WHEAT PROTEIN RESPONSE TO NITROGEN
2014 WHEAT PROTEIN RESPONSE TO NITROGEN Aron Wlters, Coopertor Ryford Schulze, Coopertor Dniel Hthcot, Extension Progrm Specilist Dr. Clrk Neely, Extension Stte Smll Grins Specilist Ryn Collett, Extension
More informationAnnouncements. CS 188: Artificial Intelligence Spring Today. P4: Ghostbusters. Exact Inference in DBNs. Dynamic Bayes Nets (DBNs)
CS 188: Artificil Intelligence Spring 2010 Lecture 21: DBNs, Viteri, Speech Recognition 4/8/2010 Written 6 due tonight Project 4 up! Due 4/15 strt erly! Announcements Course contest updte Plnning to post
More informationUnit 2: Right Triangle Trigonometry RIGHT TRIANGLE RELATIONSHIPS
Unit 2: Right Triangle Trigonometry This unit investigates the properties of right triangles. The trigonometric ratios sine, cosine, and tangent along with the Pythagorean Theorem are used to solve right
More information1 What is Trigonometry? Finding a side Finding a side (harder) Finding an angle Opposite Hypotenuse.
Trigonometry (9) Contents 1 What is Trigonometry? 1 1.1 Finding a side................................... 2 1.2 Finding a side (harder).............................. 2 1.3 Finding an angle.................................
More informationLesson 8: Application Technology
The type of ppliction equipment used must suit the type of ppliction. In this module, you ll lern the prts of the most common types of ppliction equipment used by ssistnt pplictors, s well s how to properly
More informationUnit 6  Quiz 1. Look at the pictures and write the missing letters. (5x2=10)
Unit 6  Quiz Nme & Surnme :... Stuent ID Numer :... Shool :... Clss :... Dte :... Result :... A Look t the pitures n write the missing letters. (x=0 i _ i _ r _ h g _ v kk _ e _ g B Mth the wors with
More informationFREEWAY SYSTEM PROBLEMS AND DEFICIENCIES: PHYSICAL DESIGN, TRAFFIC SAFETY, AND TRAFFIC CONGESTION
Finl Drft s Approved by Technicl Subcommittee SEWRPC Plnning Report No. 7 A REGIONAL FREEWAY RECONSTRUCTION SYSTEM PLAN FOR SOUTHEASTERN WISCONSIN Chpter V FREEWAY SYSTEM PROBLEMS AND DEFICIENCIES: PHYSICAL
More informationANATOMY OF A TRIPOD: Friction Lock (column) 8 Boots / Footwear. Friction Lock (Leg)
IMPORTANT DO NOT USE YOUR TRIPOD UNTIL YOU HAVE READ THESE INSTRUCTIONS. NO WARRANTY CLAIM WILL BE ENTERED INTO FOR THE MISUSE OR INCORRECT USE OF THESE PRODUCTS. ANATOMY OF A TRIPOD: We offer the three
More informationDesign and Calibration of Submerged Open Channel Flow Measurement Structures: Part 3  Cutthroat Flumes
Uth Stte University DigitlCommons@USU Reports Uth Wter Reserch Lbortory Jnury 1967 Design nd Clibrtion of Submerged Open Chnnel Flow Mesurement Structures: Prt 3  Cutthrot Flumes Gylord V. Skogerboe M.
More informationActivity Guide for Daisies, Brownies, and Juniors
ctivity for Dies, Brownies, Juniors World Thinking Dy I n Girl Scouts, you re prt specil group girls tht stretches cross world. On Februry 22 ech yer, Girl Scouts Girl s from 150 countries celebrte World
More informationAn Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. The tennis serve technology based on the AHP evaluation of consistency check
[Type text] [Type text] [Type text] ISSN : 0977 Volume 0 Issue 0 BioTechnology 0 An Indin Journl FULL PAPER BTAIJ, 0(0), 0 [7] The tennis serve technology bsed on the AHP evlution of consistency check
More informationAHPbased tennis service technical evaluation consistency test
Avilble online.jocpr.com Journl of Chemicl nd Phrmceuticl Reserch, 0, ():779 Reserch Article ISSN : 09778 CODEN(USA) : JCPRC AHPbsed tennis service technicl evlution consistency test Mio Zhng Deprtment
More informationI can add vectors together. IMPORTANT VOCABULARY
PreAP Geometry Chapter 9 Test Review Standards/Goals: G.SRT.7./ H.1.b.: I can find the sine, cosine and tangent ratios of acute angles given the side lengths of right triangles. G.SRT.8/ H.1.c.: I can
More informationTitle: Direction and Displacement
Title: Direction and Displacement Subject: Mathematics Grade Level: 10 th 12 th Rational or Purpose: This activity will explore students knowledge on directionality and displacement. With the use angle
More information* LANDING ROLLED CURB SIDEWALK RAMP TYPE R (ROLLED SIDES) * LANDING ** RAMP FULL CURB HEIGHT MAY BE REDUCED TO ACCOMMODATE MAXIMUM SIDE FLARE SLOPE
* MXIMUM LNDING IS 2.0% IN ECH DIRECTION OF TRVEL. LNDING MINIMUM DIMENSIONS 5' x 5'. ** MXIMUM RMP CROSS IS 2.0%, RUNNING 5%  7% (8.3% MXIMUM). "NONWLKING" RE * LNDING 24" CROSS FULL WIDTH PERMNENT
More information8.3 Trigonometric RatiosTangent. Geometry Mr. Peebles Spring 2013
8.3 Trigonometric RatiosTangent Geometry Mr. Peebles Spring 2013 Bell Ringer 3 5 Bell Ringer a. 3 5 3 5 = 3 5 5 5 Multiply the numerator and denominator by 5 so the denominator becomes a whole number.
More informationChapter 7. Right Triangles and Trigonometry
Chapter 7 Right Triangles and Trigonometry 4 16 25 100 144 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 20 32 = = = 4 *2 = = = 75 = = 40 = = 7.1 Apply the Pythagorean Theorem Objective:
More informationMORE TRIGONOMETRY
MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram
More informationSAMPLE EVALUATION ONLY
mesurement nd geometry topic 15 Pythgors theorem 15.1 Overview Why lern this? Pythgors ws fmous mthemtiin who lived out 2500 yers go. He is redited with eing the fi rst person to prove tht in ny rightngled
More information*Definition of Cosine
Vetors  Unit 3.3A  Problem 3.5A 3 49 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? ()
More informationA2.A.73: Law of Sines 4: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines
A2.A.73: Law of Sines 4: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines 1 In the accompanying diagram of ABC, m A = 65, m B = 70, and the side opposite vertex B is 7.
More informationSection 8: Right Triangles
The following Mathematics Florida Standards will be covered in this section: MAFS.912.GCO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and HypotenuseLeg) follow from the definition
More informationEasyStart Guide Works with FitnessBoxingOnline.com and other Boxout Video & Mp3 workouts
Box Trg System EyStrt Guide Wks FnessBoxgOnle.com Box Video Mp3 wks igl Auntic Trg System lets wk 3 wys EVEN IF YOU NEVER READ INSTRUCTIONS, JUST READ THIS BIT! If re COMPLETELY NEW TO BOXING OUT OF SHAPE
More informationNick Willis Discipline: Middle distance running Specialist events: 800m and 1500m
Olympic Eduction Discipline: Middle distnce running Specilist events: 800m nd 1500m ws born in Lower Hutt (ner Wellington) in 1983.Running tlent obviously runs in the Willis fmily, s Nick nd his brother
More information