5.8. Solving Three-Dimensional Problems by Using Trigonometry. LEARN ABOUT the Math. Matt s Solution. 328 Chapter 5

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1 YOU WILL NEE dynamic geometry software (optional) Solving Tree-imensional Problems by Using Trigonometry GOL Solve tree-dimensional problems by using trigonometry. LERN OUT te Mat From point, Manny uses a clinometer to determine te angle of elevation to te top of a cliff as 38. From point, 68.5 m away from Manny, Joe estimates te angle between te base of te cliff, imself, and Manny to be 42, wile Manny estimates te angle between te base of te cliff, imself, and is friend Joe to be m? Wat is te eigt of te cliff to te nearest tent of a metre? EXMPLE 1 Solving a tree-dimensional problem by using te sine law alculate te eigt of te cliff to te nearest tent of a metre. Matt s Solution In ^: / ( ) 5 75 is in ^. In ^, I don t ave enoug information to calculate, but is also in ^. In ^, I knew two angles and a side lengt. efore I could calculate, I needed to determine /. I used te fact tat te sum of all tree interior angles is apter 5 NEL

2 1 sin 42 sin 5 sin sin sin 75 3 sin sin sin 75 5 sin sin 75 Using ^ and te value of /, I used te sine law to calculate. To solve for, I multiplied bot sides of te equation by sin 42. I used my calculator to evaluate m tan 38 5 tan tan m 8 Te eigt of te cliff is about 37.1 m. Ten I used ^ to calculate. I knew tat ^ is a rigt triangle and tat is opposite te 38 angle wile is adjacent to it. So I used tangent. To evaluate, I multiplied bot sides of te equation by Reflecting. Was te given diagram necessary to elp Matt solve te problem? Eplain.. Wy did Matt begin working wit ^ instead of ^?. Wat strategies migt Matt use to ceck weter is answer is reasonable? NEL Trigonometric Ratios 329

3 PPLY te Mat EXMPLE 2 Solving a tree-dimensional problem by using te sine law Emma is on a 50 m ig bridge and sees two boats ancored below. From er position, boat as a bearing of 230 and boat as a bearing of 120. Emma estimates te angles of depression to be 38 for boat and 35 for boat. How far apart are te boats to te nearest metre? E 50 m 120 N q Q 230 Kelly s Solution In ^ Q: /Q In ^ Q, I knew tat te value of /Q is equal to te difference of te bearings of boats and. So I subtracted 120 from b E m Q 110 a q In ^ Q, I knew only one angle and no side lengts. In order to calculate q, I needed to determine Q (side b) and Q (side a) first. I drew a sketc tat included te angles of depression. I used tese angles to determine te values of /EQ and /EQ. /EQ 5 38 E /EQ 5 35 Te angle of depression to is measured from a line parallel to Q. So /EQ is equal to 38. Using te same reasoning, I determined tat /EQ is equal to m I included te values of /EQ and /EQ in my sketc. 38 b Q 110 q 35 a 330 apter 5 NEL

4 In ^EQ: tan b b 5 50 tan 38 b m q 5! q m In ^EQ: tan a q 2 5 b 2 1 a 2 2 2ba cos 110 q 2 5 (64.0) 2 1 (71.4) 2 2 2(64.0)(71.4)cos 110 Te boats are about 111 m apart. a 5 50 tan 35 a m Since ^ EQ and ^ EQ are rigt triangles, I epressed Q in terms of tan 38 and Q in terms of tan 35. Ten I solved for b and a. In ^ Q, I now knew two side lengts and te angle between tose sides. So I used te cosine law to calculate q. I substituted te values of b and a into te equation and evaluated q. In Summary Key Ideas Tree-dimensional problems involving triangles can be solved using some combination of tese approaces: trigonometric ratios te Pytagorean teorem te sine law te cosine law Te approac you use depends on te given information and wat you are required to find. Need to Know Wen solving problems, always start wit a sketc of te given information. etermine any unknown angles by using any geometric facts tat apply, suc as facts about parallel lines, interior angles in a triangle, and so on. Revise your sketc so tat it includes any new information tat you determined. Ten use trigonometry to solve te original problem. In rigt triangles, use te primary or reciprocal trigonometric ratios. In all oter triangles, use te sine law and/or te cosine law. Given Information Required to Find Use SS angle sine law S or S side sine law SS side cosine law SSS side cosine law NEL Trigonometric Ratios 331

5 35 45 m 1 m m 4.0 m 0.5 m HEK Your Understanding 1. Morana is trolling for salmon in Lake Ontario. Se sets te fising rod so tat its tip is 1 m above water and te line forms an angle of 35 wit te water s surface. Se knows tat tere are fis at a dept of 45 m. escribe te steps you would use to calculate te lengt of line se must let out. 2. Jos is building a garden sed tat is 4.0 m wide. Te two sides of te roof are equal in lengt and must meet at an angle of 80. Tere will be a 0.5 m overang on eac side of te sed. Jos wants to determine te lengt of eac side of te roof. a) Sould e use te sine law or te cosine law? Eplain. b) How could Jos use te primary trigonometric ratios to calculate? Eplain. PRTISING 3. etermine te value of to te nearest centimetre and u to te nearest K degree. Eplain your reasoning for eac step of your solution. a) c) E cm 115 F G cm E b) d) in cm cm 15 cm cm s a project, a group of students was asked to determine te altitude,, of a promotional blimp. Te students measurements are sown in te sketc at te left. a) etermine to te nearest tent of a metre. Eplain eac of your steps. b) Is tere anoter way to solve tis problem? Eplain m 332 apter 5 NEL

6 5. Wile Travis and ob were flying a ot-air balloon from eamsville to Vineland in soutwestern Ontario, tey decided to calculate te straigt-line distance, to te nearest metre, between te two towns. From an altitude of 226 m, tey simultaneously measured te angle of depression to eamsville as 2 and to Vineland as 3. Tey measured te angle between te lines of sigt to te two towns as 80. Is tere enoug information to calculate te distance between te two towns? Justify your reasoning wit calculations. 6. Te observation deck of te Skylon Tower in Niagara Falls, Ontario, is 166 m above te Niagara River. tourist in te observation deck notices two boats on te water. From te tourist s position, te bearing of boat is 180 at an angle of depression of 40 te bearing of boat is 250 at an angle of depression of 34 alculate te distance between te two boats to te nearest metre. 7. Suppose Romeo is serenading Juliet wile se is on er balcony. Romeo is facing nort and sees te balcony at an angle of elevation of 20. Paris, Juliet s oter suitor, is observing te situation and is facing west. Paris sees te balcony at an angle of elevation of 18. Romeo and Paris are 100 m apart as sown. etermine te eigt of Juliet s balcony above te ground, to te nearest metre. J R 100 m P 8. coast guard elicopter overs between an island and a damaged sailboat. From te island, te angle of elevation to te elicopter is 73. From te elicopter, te island and te sailboat are 40 apart. police rescue boat eading toward te sailboat is 800 m away from te scene of te accident. From tis position, te angle between te island and te sailboat is 35. t te same moment, an observer on te island notices tat te sailboat and police rescue boat are 68 apart. Eplain ow you would calculate te straigt-line distance, to te nearest metre, from te elicopter to te sailboat. Justify your reasoning wit calculations. NEL Trigonometric Ratios 333

7 9. rit and Tara are standing 13.5 m apart on a dock wen tey observe a sailboat moving parallel to te dock. Wen te boat is equidistant between bot girls, te angle of elevation to te top of its 8.0 m mast is 51 for bot observers. escribe ow you would calculate te angle, to te nearest degree, between Tara and te boat as viewed from rit s position. Justify your reasoning wit calculations. 10. In setting up for an outdoor concert, a stage platform as been dismantled T into tree triangular pieces as sown. G J 3.9 m 2.7 m 4.7 m m 4.5 m m I 4.5 m H L K Tere are tree veicles available to transport te pieces. In order to prevent damaging te platform, eac piece must fit eactly inside te veicle. Eplain ow you would matc eac piece of te platform to te best-suited veicle. Justify your reasoning wit calculations. 2.1 m 2.1 m 1.6 m 6.0 m 2.6 m 4.0 m 2.5 m 4.5 m 1.8 m m 11. ert wants to calculate te eigt of a tree on te opposite bank of a river. To do tis, e lays out a baseline 80 m long and measures te angles as sown at te left. Te angle of elevation from to te top of te tree is 28. Eplain if tis information elps ert to calculate te eigt of te tree to te nearest metre. Justify your reasoning wit calculations. 12. andra s omework question reads like tis: ill and ris live at different intersections on te same street, wic runs nort to sout. Wen bot of tem stand at teir front doors, tey see a ot-air balloon toward te east at angles of elevation of 41 and 55, respectively. alculate te distance between te two friends. a) andra says se doesn t ave enoug information to answer te question. Evaluate andra s statement. Justify your reasoning wit calculations. b) Wat additional information, if any, would you need to solve te problem? Justify your answer. 334 apter 5 NEL

8 Etending 13. Two roads intersect at 34. Two cars leave te intersection on different roads at speeds of 80 km/ and 100 km/. fter 2, a traffic elicopter tat is above and between te two cars takes readings on tem. Te angle of depression to te slower car is 20, and te straigt-line distance from te elicopter to tat car is 100 km. ssume tat bot cars are travelling at constant speed. a) alculate te straigt-line distance, to te nearest kilometre, from te elicopter to te faster car. Eplain your reasoning for eac step of your solution. b) etermine te altitude of te elicopter to te nearest kilometre. 14. Simone is facing nort at te entrance of a tunnel troug a mountain. Se notices tat a 1515 m ig mountain in te distance as a bearing of 270 and its peak appears at an angle of elevation of 35. fter se eits te tunnel, te same mountain as a bearing of 258 and its peak appears at an angle of elevation of 31. ssuming tat te tunnel is perfectly level and straigt, ow long is it to te nearest metre? 15. n airport radar operator locates two planes flying toward te airport. Te first plane, P, is 120 km from te airport,, at a bearing of 70 and wit an altitude of 2.7 km. Te oter plane, Q, is 180 km away on a bearing of 125 and wit an altitude of 1.8 km. alculate te distance between te two planes to te nearest tent of a kilometre. 16. Mario is standing at ground level eactly at te corner were two eterior walls of is apartment building meet. From Mario s position, is apartment window on te nort side of te building appears 44.5 m away at an angle of elevation of 55. Mario notices tat is friend Tomas s window on te west side of te building appears 71.0 m away at an angle of elevation of 34. a) If a rope were pulled taut from one window to te oter, around te outside of te building, ow long, to te nearest tent of a metre, would te rope need to be? Eplain your reasoning. b) Wat is te straigt-line distance troug te building between te two windows? Round your answer to te nearest tent of a metre. NEL Trigonometric Ratios 335