CHAPTER MMS10 L1.qxp 1/21/10 10:34 AM Page 1 2

1y ago
11 Views
2 Downloads
1.28 MB
66 Pages
Last View : 18d ago
Last Download : 2m ago
Upload by : Audrey Hope
Transcription

1/21/1010:34 AMPage 1CHAPTERTrigonometryWhat You’ll Learn Determine the measure of an acute angle in a right triangle using thelengths of two sides. Determine the length of a side in a right triangle using the length of anotherside and the measure of an acute angle.AFT Solve problems that involve more than one right triangle.Why It’s ImportantTrigonometric ratios are used by: surveyors, to determine the distance across a river or a very busy streetR2 pilots, to determine flight paths and measure crosswinds forestry technicians, to calculate the heights of treesDMMS10 L1.qxpKey Wordstangent ratioangle of inclinationindirect measurementsine ratiocosine ratioangle of elevationangle of depression1

MMS10 L1.qxp1/21/1010:34 AMPage 22.1 Skill BuilderSimilar TrianglesSimilar triangles have: the measures of matching angles equal OR the ratios of matching sides equalE12 cmPSQ30 110 A5 cm8 cm6 cmF4 cmTU 2 cmR40 These triangles are not similar because theratios of matching sides are different.PQ 12 2.4ST5QR6Compare the longest sides, 3compare the shortest sides,TU 2then compare the thirdRP 8 2pair of sides.US 440 110 C30 BDAF A D 40 B E 30 C F 110 TThese triangles are similar becausematching angles are equal.CheckR1. Which triangles in each pair are similar?a)B1.0 cmDKMYE3 cm4 cm3.0 cm5.0 cmb)5 cm3.5 cm4.0 cmF8 cm10 cmGWJD2.0 cmCXCompare the ratios of matching sides.DB5.01.4 3.5MJCDKMBC 2.01.04.02 1.3 JK3.0are not similar.The triangles26 cmCompare the ratios of matching sides.5FG0.5 XY10EFWXGE 3 460.5 0.5 YW8areThe trianglessimilar.

1/21/1010:34 AMPage 32.1 The Tangent RatioFOCUS Use the tangent ratio to find an angle measure.The Tangent RatioAAn acute angle isless than 90 .side adjacentto ABside opposite ACFinding the Tangent RatioFind the tangent ratio for G.E18FGSolutionR10AFExample 1Draw an arc at G.G is EF.The side opposite GThe side adjacent to G is GE.length of side opposite Gtan G length of side adjacent to GEFtan G GE18tan G 10tan G 1.8length of side opposite Alength of side adjacent to ATIf A is an acute angle in a right triangle, then tan A DMMS10 L1.qxpEopposite18F10adjacentGSubstitute: EF 18 and GE 10The side opposite the right angleis always the hypotenuse.3

MMS10 L1.qxp1/21/1010:34 AMPage 4Check1. a) Find tan P.QR .The side opposite P isRP .The side adjacent to P isb) Find tan Q.PThe side opposite Q isRP .8QThesideadjacentlength of sideopposite P10tan P QR .to Q isRlength of sideadjacent to Plength of side opposite Qtan Q QRlength of side adjacent to Qtan P RPRP10tan Q tan P QR8tan P 1.25tan Q 810Ttan Q 0.8To find the measure of an angle, use the tan 1 key on a scientific calculator.Using the Tangent Ratio to Find the Measure of an AngleAFExample 2Find the measure of AA to the nearest degree.C16BR7SolutionDAThe side opposite AA is BC.The side adjacent to A is AB.length of side opposite Atan A length of side adjacent to AIf you are using a differentBCcalculator, consult the user’stan A Substitute: BC 16 and AB 7ABmanual.16tan A 7To find A using a TI-30XIIS calculator, enter:tan -1 (16/7)66.37062227%@.3W4E , A 66 4

1/21/1010:34 AMPage 5Check1. Find the measure of each indicated angle to the nearest degree.a) FF10G13HThe side opposite F isGH .FG .The side adjacent to F isTlength of sideopposite Flength of sideadjacent to Ftan F GHFGtan F 1310tan F 1.3AFtan F Use a calculator.R(1.3) F tan 152 F b) ECDMMS10 L1.qxp5DThe side opposite E isCD .The side adjacent to E isDE .tan E length of side opposite Elength of side adjacent to Etan E CDDEtan E 599E E 5a b9tan 1 E 29 5

MMS10 L1.qxp1/21/1010:34 AMExample 3Page 6Using the Tangent Ratio to Find an Angle of InclinationA guy wire is fastened to a cell-phone tower 8.5 mabove the ground. The wire is anchored to the ground14.0 m from the base of the tower. What angle, to thenearest degree, does the wire make with the ground?The angle the wire makeswith the ground is called theangle of inclination.SolutionADraw a diagram.The angle the wire makes with the ground is B.To find B, use the tangent ratio.Btan B length of side opposite Blength of side adjacent to B8.5 m14.0 mAssume the tower isperpendicular to theground.CThe side opposite BB is CA.The side adjacent to BB is BC.CASubstitute: CA 8.5 and BC 14.0BC8.5tan B Use a calculator.14.0 B 31 The angle between the ground and the wire is about 31 .AFTtan B RCheckFD1. A ladder leans against a house. The top of the ladder is 2.4 m above the ground.Its base is 0.9 m from the wall. What angle, to the nearest degree, does theladder make with the ground?Label the given triangle FGH.Label G where the ladder meets the ground.Label F where it meets the wall.We want to find the measure of G.2.4 mGThe side opposite G is .HFThe side adjacent to G is .GHtan G HFGHtan G 2.40.9tan G 69 G The angle between the ground and the ladder is about69 .6H0.9 moppositeadjacent

1/21/1010:34 AMPage 7Practice1. Label the hypotenuse, opposite, and adjacent sides of each right trianglein relation to the given angle.a) Hb) djacentNH2. Find the tangent ratio for each indicated angle. Leave the ratio in fraction form.a)b)VTZ71615AFXU3Ytan Y RZX .The side opposite Y is.XY .The side adjacent to Y is . Ylength of sideoppositelength of sideadjacent to YWUVThe side opposite W is . VWThe side adjacent to W is .tan W length of side opposite Wlength of side adjacent to Wtan Y ZXXYtan W UVVWtan Y 73tan W 1615DMMS10 L1.qxp3. Find the measure of A for each value of tan A. Give your answer to the nearest degree.5a) tan A 0.5b) tan A 60.5 ) A tan 1(27 A Use a calculator.5tan 1a b6 A 40 A 7

MMS10 L1.qxp1/21/105:24 PMPage 84. Find the measure of B to the nearest degree.B13CCDThe side opposite B is.BCThe side adjacent to B is.12tan B Dlength of sideopposite Blength of sideadjacent to Btan B CDBCtan B 121343 B M5. A telephone pole is supported by a wire, as shown.What angle, to the nearest degree, does the wire make with the ground?We want to find the measure of N.Use the tangent ratio.length of sideadjacent to NNPMNPNtan N8mPR188Ntan 66 N Tlength of sideopposite NAFNtan 18 mD66 .The angle between the ground and the wire is about6. Victor is building a wheelchair ramp to an entranceway that is 3 m above the sidewalk.The ramp will cover a horizontal distance of 50 m. What angle, to the nearest degree,will the ramp make with the ground?R3mS50 mQWe want to find the measure of Q.Use the tangent ratio.tan Q ⴝlength of side opposite Qlength of side adjacent to Qtan Q ⴝRSSQ350 Q 3 3 .The angle between the ground and the ramp is abouttan Q ⴝ8TEACHER NOTENext Steps: Havestudents completequestions 6, 8, 10, 14,and 15 on pages 75and 76 of the StudentText.For studentsexperiencing success,introduce Example 4,on page 74 of theStudent Text, andassign Practicequestions 13, 19,and 20.

1/21/1010:35 AMPage 92.2 Skill BuilderSolving EquationsInverse operations “undo” each other’s results.Multiplication and division are inverse operations.We can use inverse operations to solve some equations.a 4:5To solve36 9:ba 45a5 5 45a 20Undo the division.Multiply each side by 5.36 b 9bUndo the division.Multiply each side by b.b 36 9bRecall: b 9 9bChecka)R1. Solve each equation.Undo the multiplication.Divide each side by 9.AF36 9b 994 bTTo solvea 57Multiply each side by7 .a 57 7735a c8Multiply each side by8 .c8 9 8 872 cb) 9 DMMS10 L2.qxpc) 13 156fd)Multiply each side byf . 13 ff156f13f 15613Divide each side by .15 6b15b b b 615 6b6b15 662.5 b15613f 1313 f129

MMS10 L2.qxp1/21/1010:36 AMPage 102.2 Using the Tangent Ratio toCalculate LengthsFOCUS Use the tangent ratio to calculate lengths.When we know the measure of an acute angle and the length of a leg of aright triangle, we can use the tangent ratio to find the length of the other leg.Example 1Finding the Length of an Opposite SideFind the length of BC to the nearest tenth of a centimetre.23.0 cm28 CATBSolutionAopposite AAlength of side opposite Alength of side adjacent to ABCCABCtan 28 23Solve the equation for BC.Multiply each side by 23.BC23 tan 28 23 2323 tan 28 BCBSubstitute: A 28 and CA 23Dtan A CRtan A adjacent to A23.0 cm28 AFWe know the measure of A.BC is the side opposite A.CA is the side adjacent to A.Use the tangent ratio to write an equation.BC 12.2293 BC is about 12.2 cm long.10Use a calculator. Enter: / 0 V @ / 5 E 23*tan(28)12.22931693

1/21/1010:36 AMPage 11Check1. Find the length of each indicated side to the nearest tenth of a centimetre.The given angle is F.DEis the side opposite F.is the side adjacent to F.EFa) Side EDF59 8 cmDEtan F sideopposite Fsideadjacent to FDEtan F EFDEtan859 Solve the equation for DE.Multiply each side by.8TDE859 tan 88AF tan 859 DEDE 13.3142 13.3 cm long.DE is aboutb) Side HJRJ13 cmH34 KDMMS10 L2.qxpThe given angle is K .HJis the side opposite K . K .JKis the side adjacent toside opposite KKside adjacent to Ktan HJtan KJKHJ13tan 34 13 tan 34 13 HJ1313 tan 34 HJHJ 8.7686 8.8 cm long.HJ is about11

MMS10 L2.qxp1/21/1010:36 AMExample 2Page 12Finding the Length of an Adjacent SideFind the length of PQ to the nearest tenth of a centimetre.Q5.0 cmR35 PSolutionUse the tangent ratio to write an equation.side opposite Ptan P side adjacent to Ptan 35 5PQSubstitute: QR 5 and PP 35 Solve the equation for PQ.Multiply each side by PQ.PQ tan 35 PQ 5PQPQ tan 35 5Divide each side by tan 35 .R5PQ tan 35 tan 35 tan 35 5tan 35 35 PQ 7.1407 DPQ So, PQ is about 7.1 cm long.12TQRPQAFtan P We know P 35 .QR is opposite P.PQ is adjacent P.Use a calculator.

1/21/1010:36 AMPage 13Check1. Find the length of TU to the nearest tenth of a centimetre.S4.0 cmU39 TThe given angle is T.US is the side opposite T.TU is the side adjacent to T.sideopposite Tsideadjacent to TUStan T TU4TU39 tan4TU tan39 tan 39 TU tan 39 Divide each side by .4tan 39 RT U tan39 TU .Multiply each side byAF4TU TU39 TU tanSubstitute: T T 39 andUS 4Ttan T 4tan 39 39 Use a calculator.4.9395 TU DMMS10 L2.qxpTU is about4.9 cm long.13

MMS10 L2.qxp1/21/1010:36 AMExample 3Page 14Using the Tangent Ratio to Solve a ProblemA wire supports a flagpole. The angle between the wire and the level ground is 73 . The wireis anchored to the ground 10 m from the base of the pole. How high up the pole does thewire reach? Give the answer to the nearest tenth of a metre.SolutionSketch and label a diagram.Assume the flagpole meets the ground at a right angle.BThe given angle is A. We want to find the length of BC.tan A side opposite Aside adjacent to ABCCAA73 C10.0 mSubstitute: AA 73 and CA 10Solve the equation for BC.Multiply each side by 10.BC1010 tan 73 BCtan 73 AFtan A TBC is opposite A.A.CA is adjacent to A.A.Use a calculator.RBC 32.7085 CheckDThe wire reaches the flagpole at a height of about 32.7 m.1. A ladder leans on a wall, as shown. How far up the wall doesthe ladder reach? Give your answer to the nearest tenthof a metre.The given angle is F.DEWe want to find the length of .D67 EF 1.5 m14

1/21/1010:36 AMPage 15sideopposite Ftan F tan F sideadjacent to FDEEFDE1.567 tanSubstitute: F 67 andEF 1.51.5 .Multiply each side by67 1.5 tanDE3.5337 DE The ladder reaches the wall at a height of about3.5 m .TPractice1. Find the length of the side opposite the given angle to the nearest tenth of a centimetre.a)b)AFGZX 57 F5 cm31 6 cmHRThe given angle is F.GH .The side opposite F is HF .The side adjacent to F istan F sideopposite FDMMS10 L2.qxpsideadjacent to FGHtan F HFGH31 tan6GH6 tan 6631 GH31 6 tan3.6051 GH 3.6 cmGH is aboutlong.YXThe given angle is .X is .YZThe side opposite XYXThe side adjacent tois .tan Xsideopposite Xsideadjacent to XYZXYYZtan 57 5tan X 5 tan 57 5 5 tan 57 YZYZ5YZ 7.6993 YZ is about7.7 cmlong.15

MMS10 L2.qxp1/21/1010:36 AMPage 162. Find the length of CD to the nearest tenth of a centimetre.DThe given angle is C.DE .The side opposite C isCD .The side adjacent to C is16 cmE44 tan C Csideopposite Csideadjacent to CDEtan C CD1644 tanCDCDMultiply each side by .CD tan 44 CD 16CDCD tan 44 16AFTtan 44 .Divide each side byCD tan 44 16 tan 44 tan 44 44 16CD tan 44 44 CD 16.5684 16.6 cm long.CD is aboutR3. Find the length of the indicated side to the nearest tenth of a centimetre.a) Side PQDPb) Side UVVQU60 32 cmT18 cm36 oppositeadjacenttan URtan R oppositeadjacentPQtan36 18PQ18 tan 36 181818 tan 36 PQPQ 13.0777 13.1 cm long.PQ is about1632UV60 tanUV tan 60 UV 32UVUV tan 60 32UV tan 60 32 tan 60 tan 60 UV 18.4752 UV is about18.5 cm long.

1/21/1010:36 AMPage 174. This diagram shows an awning over the windowof a house. Find the height of the awning, GH,to the nearest tenth of a metre.tan H tan 32 H32 oppositeadjacentG 1.6 m1.6GHGH tan 32 GH 1.6GHJwindowGH tan 32 1.61.6GH tan 32 tan32 tan 32 GH 2.5605 2.6 m .The height of the awning is aboutT5. A rope supports a tent. The angle between the rope and the level ground is 59 . The rope isattached to the ground 1.2 m from the base of the tent. At what height above the ground isthe rope attached to the tent? Give your answer to the nearest tenth of a metre.59 1.2 mCRAAFBWe want to find the length of BC.oppositetan A adjacentBCtan A ACDMMS10 L2.qxptan 59 BC1.2BC1.2 tan 59 1.2 1.21.2 tan 59 BCBC 1.9971 The rope is attached to the tent at a height of about2.0 m .TEACHER NOTENext Steps: Havestudents completequestions 6, 8, 10, 11,and 13 on pages 82and 83 of the StudentText.17

MMS10 L3.qxp1/21/1010:37 AMPage 182.3 Math Lab: Measuring anInaccessible HeightFOCUS Determine a height that cannot be measured directly.When we find a length or an angle without using a measuring instrument, we areusing indirect measurement.Try ThisWork with a partner.Follow the instructions in Part A on Student Text page 85 to make a clinometer.The materials you need are listed on Student Text page 84.Record all your measurements on the diagram below.Mark a point on the ground.Measure the distance to the base of the object.TChoose a tall object; for example, a tree or a flagpole.Object:AFOne person stands at the point. He holds the clinometer, then looks at the top of the objectthrough the straw. The other person records the angle shown by the thread on the protractor.Then that person measures the height of the eyes above the ground of the person holdingthe clinometer.RSubtract the clinometer angle from 90 .This is the angle of inclination of the straw.BDClinometer angle:Height of object:Angle of inclination:ACHeight of eyes:Distance to object:18

1/21/1010:37 AMPage 19Use the tangent ratio to calculate the length of BC:BCtan A ACBCtan BC tanBC Height of object length of BC height of eyes above the groundHeight of object Height of object Change places with your partner.Repeat the activity.Does the height of your eyes affect the measurements? Explain. Sample response:TYes,the height of your eyes affects the angle of inclination of the object.Thetaller you are, the less you have to look up to the object.AFDoes the height of your eyes affect the final result? Explain. Sample response:No,a taller person will calculate a greater height above the horizontal but will have alesser eye height. But, when the two heights are added, both results should be theRsame.DPractice1. Which angle of inclination does each clinometer measure?2000100806040160 140 1200b)180 160 140018a)1210080604020MMS10 L3.qxpAngle of inclination 90 angle on clinometer 90 55 Angle of inclination 90 80 10 35 19

Page 2012008010EG1.3 m4.0 mFG30 tan 44 tan 30 ⴝ 4 a60 0FAngle on clinometer:60 Angle of inclination: 90 30 60 So, E 30 oppositetan E adjacentFG30 tanEG140 160 1802. Use the information in the diagram to findthe height of the flagpole to the nearesttenth of a metre.605:08 PM401/21/10FGb42.3094 FG T4 tan 30 ⴝ FGSo, height of flagpole FG height of eyes above groundAF 1.32.3094 3.6094 The height of the flagpole is about3.6 m .020806040D100QR10QRP1.7 m10 m10 tan 15 ⴝ QR2.6794 QR So, height of tree 2.6794 ⴙ 1.7 4.3794 4.4 m .The height of the tree is about20120oppositeadjacentQR PR15 tantan 15 ⴝ160 140R75 P is: 90 75 15 tan P 1803. Use the information in the diagram to findthe height of the tree to the nearest tenthof a metre.This diagram is not drawn to scale.TEACHER NOTENext Steps: Havestudents completequestion 1 on page 86of the Student Text.20MMS10 L3.qxp

MMS10 L3.qxp1/21/1010:37 AMPage 21Can you CHECKPOINT 1 use the tangent ratio to find an angle measure? use the tangent ratio to calculate a length? use the tangent ratio to solve a problem?1. Find the tangent ratio for each indicated angle. Leave the ratio in fraction form.a) B11b)CE22157DFABCThe side opposite A is .tan D sideopposite Asideadjacent to Atan D AFtan A tan D BCtan A AB11tan A 7side opposite Dside adjacent to DTABThe side adjacent to A is .EFDE1522a)G8F17R2. Find the measure of each indicated angle to the nearest degree.D2.1b)K7JHThe side opposite HH is .FG22MThe side adjacent to H isGH .tan H sideopposite Hsideadjacent to HFGtan H GH8tan H 17 H 25 side opposite Kside adjacent to KK tanMJK tanJK22K tan7 K 72 21

MMS10 L3.qxp10:37 AMPage 223. Find the length of each indicated side to the nearest tenth of a centimetre.a) Side STb) Side PQT24 cmPR21 cm38 U68 Qside opposite QQside adjacent to Qtan Ssideopposite Usideadjacent to USTtan U TURPPQtan 68 24PQPQ tan 68 PQ ST38 21tanST21 tan 38 21 2121 tan 38 ST24PQPQ tan 68 24PQ tan 68 68 24 tan 68 tan 68 68 PQ 9.6966 PQ is about9.7 cm long.AF16.4069 ST 16.4 cm long.ST is abouttan Q Ttan U R4. Margy is building a support brace to reach the top of a wall, as shown.How far from the wall should the brace be anchored to the ground?Give your answer to the nearest tenth of a metre.We want to find the length of AB.C3.5 mThe side opposite AA is BC.The side adjacent to A A is AB.side opposite Atan A side adjacent to ABCtan A AB3.5tan 70 AB3.5AB tan 70 AB ABAB tan 70 3.5AB tan 70 3.5 tan70 tan 70 3.5AB tan 70 AB 1.2738 about 1.3 m from the wall.The brace should be anchored to the groundD2.21/21/1022B70 ATEACHER NOTENext Steps: Havestudents completequestions 3 and 5on page 88 of theStudent Text.

1/21/1010:38 AMPage 232.4 Skill BuilderSum of the Angles in a TriangleIn any triangle, the sum of theangle measures is 180 .So, to find an unknown angle measure: start with 180 subtract the known measuresIn any right triangle, the sum of themeasures of the acute angles is 90 .So, to find the measure of an acute angle: start with 90 subtract the known acute angleDA40 E71 64 BFC E 90 40 EE 50 AFT C 180 71 64 C 45 Check1. Find the measure of the third angle.a)46 RGMb)H30 DMMS10 L4.qxpO59 76 JN30 46 H 180 104 H M 180 ⴚ 76 ⴚ 59 45 M 2. Find the measure of the third angle.Ta)Qb)56 49 SU U 90 56 34 U RP P 90 ⴚ 49 41 P 23

MMS10 L4.qxp1/21/1010:39 AMPage 242.4 The Sine and Cosine RatiosFOCUS Use the sine and cosine ratios to determine angle measures.CThe Sine and Cosine RatiosIn a right triangle, if A is an acute angle, thenlength of side opposite Asin A length of hypotenusecos A side opposite Alength of side adjacent to Alength of hypotenuseExample 1hypotenuseABside adjacent to AFinding the Sine and Cosine of an AngleATFind sin B and cos B to the nearest hundredth.26 cm10 cmCSolutionlength of side opposite Blength of hypotenuseACAB24sin B 26sin B 0.9230 sin B 0.92sin B ASubstitute: AC 24 and AB 26BC is the side adjacent to B.AB is the hypotenuse.cos B cos B cos B cos B cos B 24hypotenuse26 cmB24 cmside opposite BDsin B RAC is the side opposite B.AB is the hypotenuse.The nearest hundredthmeans two decimal places.AFB24 cmlength of side adjacent to Blength of hypotenuseBCSubstitute: BC 10 and AB 26AB10260.3846 0.3810 cmside adjacent to BC

1/21/1010:39 AMPage 25Check1. Find sin D and cos D to the nearest hundredth.sin D side opposite Dhypotenusecos D Dside adjacent to DhypotenuseEFsin D DEFDcos D DE4.08.5sin D 7.58.5cos D 0.4705 sin D 0.8823 cos D sin D 0.47cos D 0.887.5 cmF8.5 cm4.0 cmEUsing the Sine or Cosine Ratio toFind the Measure of an AngleAFExample 2TTo find the measure of an angle, use the sin 1 or cos 1 key on a scientific calculator.Find the measures of B and Dto the nearest degree.4.6 cmBRSolutionCD8.7 cmFind the measure of BB first.BC is adjacent to B.B. BD is the hypotenuse.DMMS10 L4.qxpSo, use the cosine ratio to write an equation.cos B side adjacent to BBhypotenusecos B BCBDcos B 4.68.7We could use the sine ratioto find the measure of Dfirst.Substitute: BC 4.6 and BD 8.7To find B using a TI-30XIIS calculator, enter:%?183W584E B 58 cos -1 (4.6/8.7)58.07993367Since the sum of the acute angles in a right triangle is 90 , D 90 BSo, D 90 58 D 32 Since the measure of Bis an estimate, so is themeasure of D.25

MMS10 L4.qxp1/21/1010:39 AMPage 26Check1. Find the measure of each acute angle to the nearest degree.a) U5 cmFind the measure of U first.side adjacent to U.UV is thehypotenuse .UW is theSo, use the cosine ratio to write an equation.V9 cmside adjacent to Ucos U hypotenuseWUVcos U UW5cos U 956 U b)10.8 cmAFTUSo, W 90 U56 W 90 34 W UU first.Find the measure of side opposite U .ST is thehypotenuseSU is the.sineSo, use the ratio to write an equation.TSside opposite UsinhypotenuseU R18.0 cmDUSTsinUSU 10.8sin18.0U 37 U USo, S 90 37 S 90 53 S 26

1/21/1010:39 AMExample 3Page 27Using Sine or Cosine to Solve a ProblemA storm caused a 15.3-m hydro pole to lean over. The top of the pole is now 12.0 mabove the ground. What angle does the pole make with the ground? Give the answer tothe nearest degree.SolutionDraw a diagram. AC represents the pole.CThe pole meets the ground at A.BC is the side opposite A. AC is the hypotenuse.15.3 m12.0 mSo, use the sine ratio to find A.side opposite AhypotenuseAsin A BCACSubstitute: BC 12.0 and AC 15.3Assume the ground ishorizontal.sin A 12.015.3Use a calculator.BAFTsin A A 52 CheckRSo, the hydro pole makes an angle of about 52 with the ground.1. A ladder leans on a wall as shown.What angle does the ladder make with the ground?Give your answer to the nearest degree.We want to find the measure of D.DF is.adjacent to DDDE is.the hypotenuseSo, use theratio to find D.cosineDMMS10 L4.qxpE8mD 2m FAssume the ground ishorizontal.27

MMS10 L4.qxp1/21/1010:39 AMPage 28side adjacent to DhypotenusecosD DFDED cosDF ⴝ 2 andSubstitute:DE ⴝ 82cos8D D 76 So, the ladder makes an angle of about76 with the ground.2. The string of a kite is 160 m long. The string is anchored to theground. The kite is 148 m high. What angle does the string makewith the ground? Give your answer to the nearest degree.U148 mWe want to find the measure of V.side opposite V .TU is thehypotenuseUV is the.sine ratio to write an equation.So, use thesin V ⴝTUUVsin V ⴝ148160R V 68 Tside opposite VhypotenuseAFsin V ⴝTD68 The angle the string makes with the ground is about.Practice1. Fill in the blanks.a)b) GC3B5.045DB4.81.4FThe side opposite B isCD .The side adjacent to B isBC .The hypotenuse isBD .28The side opposite B isFG .The side adjacent to B is .BFThe hypotenuse is .GB160 mV

1/21/1010:39 AMPage 292. For each triangle in question 1, find sin B and cos B as decimals.a) sin B sideopposite Bsideadjacent to Bcos B hypotenusehypotenuseCDsin B BDBCcos B BD4sin B 53cos B 5sin B 0.80.6cos B side adjacent to Bcos B hypotenuseside opposite Bb) sin B hypotenuseFGGBcos B ⴝBFGBsin B ⴝ4.85.0cos B ⴝ1.45.0Tsin B ⴝcos B 0.28AF0.96sin B 3. Find the measure of each indicated angle to the nearest degree.a) Ab)6.9 cmBCR5.5 cmXW9.9 cmY11.7 cmside opposite BB.AC is theside adjacent to YXY is the .AB is thehypotenuse .hypotenuseWY is the.sineSo, use theratio.cosineSo, use theratio.DMMS10 L4.qxpsin B ⴝside opposite BhypotenuseACAB5.5sin B ⴝ6.9 B 53 sin B ⴝcos Y ⴝside adjacent to YhypotenuseXYWY9.9cos Y ⴝ11.732 Y cos Y ⴝ29

MMS10 L4.qxp1/21/104:18 PMPage 304. A firefighter rests a 15.6-m ladder against a building, as shown.What angle does the ladder make with the ground?Give your answer to the nearest degree.G15.6 mWe want to find the measure of H.side adjacent to H .FH is thehypotenuseGH is the.So, use theratio.cosineF8.5 mHside adjacent to Hcos H hypotenuseFHGHH cos8.5cos H 15.6T H 57 57 .The angle the ladder makes with the ground is aboutN1.6 mR4.5 mAF5. A loading ramp is 4.5 m long. The top of the ramp has height 1.6 m.What angle does the ramp make with the ground?Give your answer to the nearest degree.PMD M.We want to find the measure of NP is the side opposite M.M.MN is the hypotenuse.So, use the sine ratio.sin M ⴝside opposite Mhypotenusesin M ⴝNPMNsin M ⴝ1.64.5 M 21 The angle the ramp makes with the ground is about21 .30TEACHER NOTENext Steps: Havestudents completequestions 7, 8, 10, 11,13, and 14 on pages 95and 96 of the StudentText.

1/21/1010:40 AMPage 312.5 Using the Sine and Cosine Ratiosto Calculate LengthsFOCUS Use the sine and cosine ratios to determine lengths.To use the sine or cosine ratio to find the length of a leg, we need to know: the measure of an acute angle, and the length of the hypotenuseExample 1Using the Sine or Cosine Ratio to Find the Length of a LegFind the length of RS to the nearest tenth of a metre.R28 S9.6 mSolutionRside adjacent to Scos S hypotenusecos S RSQScos 28 RS9.69.6 cos 28 9.6 SQSubstitute: S 28 and QS 9.6Multiply both sides by 9.6.RS9.69.6 cos 28 RSRS 8.4762 RS is about 8.5 m long.adjacent to S28 9.6 mhypotenuseRThe measure of S is known.RS is the side adjacent to S.QS is the hypotenuse.So, use the cosine ratio.AFTQDMMS10 L5.qxpThe cosine ratio comparesthe adjacent side to thehypotenuse.Use a calculator.31

MMS10 L5.qxp1/21/1010:40 AMPage 32Check1. Find the length of each indicated side to the nearest tenth of a centimetre.The measure of B is known.side opposite B .AC is thehypotenuseBC is the.sine ratio.So, use thea) ACB40 15.6 cmsin B Asideopposite Bhypotenusesin B ACBCACsin 40 15.6C15.6 sin 40 15.6 AC15.6b) DEE55 19.5 cmDD32D is known.The measure of side adjacent to D .DE is thehypotenuseDF is the.cosine ratio.So, use theRFAFT15.6 sin 40 AC10.0274 AC 10.0 cm long.AC is aboutcos D side adjacent to Dhypotenusecos D DEDFDE19.519.5 cos 55 DEDE 11.1847 11.2 cm long.DE is aboutcos 55

1/21/1010:40 AMPage 33To use the sine or cosine ratio to find the length of the hypotenuse, we need to know: the measure of an acute angle, and the length of one legExample 2Using the Sine or Cosine Ratio to Find the Length of theHypotenuseFind the length of the hypotenuse to the nearest tenth of a centimetre.NThe hypotenuse is the sideopposite the right angle.9.5 cm52 PSolutionside opposite MhypotenuseNPsin M MN9.5MNMN sin 52 9.5sin 52 9.5sin 52 hypotenuse52 Mopposite M9.5 cmPMultiply both sides by MN.Divide both sides by sin 52 .MN sin 52 9.5 sin 52 sin 52 MN N M 52 and NP 9.5Substitute: MRsin M AFThe measure of M is known.NP is the side opposite M.MN is the hypotenuse.So, use the sine ratio to write an equation.TMDMMS10 L5.qxpThe sine ratio comparesthe opposite side to thehypotenuse.Use a calculator.MN 12.0556 MN is about 12.1 cm long.33

MMS10 L5.qxp1/21/104:20 PMPage 34Check1. Find the length of each hypotenuse to the nearest tenth of a centimetre.a)Ksin J sideopposite Jhypotenuse17.4 cmKMsin J JKM39 17.4JK39 sinJTheTheTheUsemeasure of J is known.KMside opposite J is:hypotenuse is:JKthe sine ratio.17.439 JK sin17.4JK sin 39 JK 27.6488 b)29 SR11.9 cmRQAFT27.6 cm long.JK is aboutDis known.The measure of Sadjacent to SS .QS is the sidehypotenuseRS is the.cosine ratio.So, use the34cos S ⴝside adjacent to Shypotenusecos S ⴝQSRScos 29 ⴝ11.9RSRS cos 29 ⴝ 11.9RS ⴝ11.9cos 29 13.6059 RS 13.6 cm long.RS is about

1/21/1010:40 AMPage 35Example 3Using Sine or Cosine to Solve a ProblemA surveyor makes the measurements shown in the diagram to find the distance between twoobservation towers on opposite sides of a river. How far apart are the towers? Give theanswer to the nearest metre.RiverTower 163 m73 Tower 2SolutionThe distance between the towers is the hypotenuse, AC.B63 m73 AFCTower 2TATower 1RThe measure of C is known.BC is the side adjacent to C.AC is the hypotenuse.So, use the cosine ratio.side adjacent to CCcos C hypotenuseBCAC63cos 73 ACAC cos 73 6363AC cos 73 AC 215.4791 The distance between the towers iscos C DMMS10 L5.qxpSubstitute: C 73 and BC 63Multiply both sides by AC.Divide both sides by cos 73 .Use a calculator.about 215 m.35

MMS10 L5.qxp1/21/1010:40 AMPage 36Check1. Sam and Sofia are building a wooden ramp for skateboarding. The height of the rampis 0.75 m. The ramp makes an angle of 8 with the ground. What length of plywood doSam and Sofia need for the top of the ramp? Give your answer to the nearest tenthof a metre.E0.75 mDF8 We want to find the length of DE.The measure of D is known.EFThe side opposite D is:DEThe hypotenuse is:So, use the sine ratio.EFsin D DETSubstitute: D D 8 and EF 0.750.75DE8 sinAFMultiply both sides byDE .8 DE sin0.750.75sin 8 RDE sin 8 .Divide both sides by5.3889 DE DSam and Sofia need about5.4 m of plywood.Practice1. Which

Find the measure of A to the nearest degree. Solution The side opposite A is BC. The side adjacent to A is AB. tan A tan A Substitute: BC 16 and AB 7 tan A To find A using a TI-30XIIS calculator, enter: . MMS10_L1.qxp 1/21/10 10:34

Related Documents:

Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .

TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26

DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .

About the husband’s secret. Dedication Epigraph Pandora Monday Chapter One Chapter Two Chapter Three Chapter Four Chapter Five Tuesday Chapter Six Chapter Seven. Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Thirteen Chapter Fourteen Chapter Fifteen Chapter Sixteen Chapter Seventeen Chapter Eighteen

18.4 35 18.5 35 I Solutions to Applying the Concepts Questions II Answers to End-of-chapter Conceptual Questions Chapter 1 37 Chapter 2 38 Chapter 3 39 Chapter 4 40 Chapter 5 43 Chapter 6 45 Chapter 7 46 Chapter 8 47 Chapter 9 50 Chapter 10 52 Chapter 11 55 Chapter 12 56 Chapter 13 57 Chapter 14 61 Chapter 15 62 Chapter 16 63 Chapter 17 65 .

HUNTER. Special thanks to Kate Cary. Contents Cover Title Page Prologue Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter

Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 . Within was a room as familiar to her as her home back in Oparium. A large desk was situated i

1-4 AutoCAD 2016 Tutorial: 2D Fundamentals Note that AutoCAD automatically assigns generic names, Drawing X, as new drawings are created. In our example, AutoCAD opened the graphics window using the default system units and assigned the drawing name Drawing1. 2. If necessary, click on the down-arrow in the Quick Access bar and select Show Menu Bar to display the AutoCAD Menu Bar. The Menu Bar .