Lesson 2 PRACTICE PROBLEMS Using Trigonometry in Any Triangle

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1 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 non-right tringles. Investigtion Prtie Prolem Options Mx Possile Points Totl Points Erned Investigtion 1: The Lw of Sines #1, 2 6 points Investigtion 2: The Lw of osines #3, 4, 5, 6 12 points Investigtion 3: Tringle Models One, Two, or None? #7, 8 8 points 6 points ** In order to ern redit for prtie prolems, LL WORK must e shown.**

2 On Your Own FOR FULL REDIT, SET-UP/WORK MUST E SHOWN ON LL PROLEMS pplitions INVESTIGTION 1 1 pilot flying due est out of Denver gets word tht mjor thunderstorm is diretly in his plne s pth. The pilot turns 35 to the left of his intended ourse nd ontinues on this new flight pth. fter voiding the worst of the storm, he turns 45 to the right of the new ourse nd flies until returning to his originl intended line of flight. The plne rehes its originl intended ourse t point 80 kilometers from the strt of the detour.. Drw sketh of this sitution.. How muh frther did the irrft trvel due to the detour? /1 2 Refer k to the Grnd nyon mpping prolem in the Think out This Sitution t the eginning of this lesson (pge 489). In ddition to lulting the distnes of prtiulr points in the Grnd nyon from tringultion points on the rim, the surveyors wnted to mp the reltive depths of these points in the nyon. Using their trnsits, they were le to mesure the ngle of depression (tht is, the ngle their line of sight mde with the horizontl) to the point of interest in the nyon.. In Prolem 4 of Investigtion 1, you found tht the distne from point on the rim to the tip of the pointed spur t ws out 1.51 miles. Suppose the surveyor mesured the ngle of depression from point to point to e 28. How fr elow point is point in vertil distne? Find the vertil distne in miles nd in feet (1 mile = 5,280 feet).. If point is 7,392 feet ove se level, how mny feet ove se level is point?. The olordo River t the ottom of the Grnd nyon is 1,850 feet ove se level. How mny feet ove the olordo River is point? Point? LESSON 2 Using Trigonometry in ny Tringle 503

3 On Your FOR OwnFULL REDIT, SET-UP/WORK MUST E SHOWN ON LL PROLEMS INVESTIGTION 2 3 Two lighthouses nd re 50 km prt. t 2.M., freighter moving prllel to line is sighted t point s shown in the digrm elow km 105. How fr is the freighter from lighthouse? From lighthouse?. t 3.M., the ngle t is 86. The ngle t is 29. How fr is the freighter from lighthouse? From lighthouse?. How fr hs the freighter moved in the hour etween 2.M. nd 3.M.? 4 The ninth hole t Duffy s Golf lu is 325 yrds down stright firwy. In his first round of golf for the seson, ndy tees off nd hooks the ll 20 to the left of the line from the tee to the hole. The ll stops 205 yrds from the tee t point P, s shown in the figure. P 205 yrds. How fr is his ll from the hole (mrked y the flg)?. To deide whih lu to use on his next shot, ndy knows he hits n verge of yrds with five iron; with four iron, he hits yrds; nd with three iron, he hits yrds. Whih of these lus would e his est hoie? 5 field is in the shpe of qudrilterl D s shown elow. Tee m m m D. Find the length D of its fourth side.. Find the mesures of the remining ngles of the field.. Find the re of the field. 504 UNIT 7 Trigonometri Methods

4 FOR FULL REDIT, SET-UP/WORK MUST E SHOWN ON LL PROLEMS On Your Own 6 side view of relining lwn hir is illustrted elow. Its funtion is sed on tringle in whih the lengths of two sides re fixed, nd the length of the third side n vry. The key tringulr omponent ( ) in the design of the lwn hir is shown in the digrm t the right. In, nd re oth 10 inhes, nd n e set to 8, 10, 12, or 14 inhes. The relining ngle for someone sitting in the hir is D. 10 in. 10 in. D. When the length, denoted, is set, the lengths of ll three sides of re fixed. How n you find m? How is m D, the mesure of the relining ngle, relted to m?. Write rule tht expresses m D in terms of.. Use the rule in Prt to find the mesure of the relining ngle for = 8, 10, 12, nd 14 inhes. Desrie how the mesure of the relining ngle hnges s hnges. 7 In the fll of 1986, three friends hllenged eh other to pumpkin-throwing ontest. The winner threw pumpkin 126 feet. This event hs sine evolved into the nnul Punkin hunkin World hmpionship in whih ir ompression nnons nd tpult mhines hurl pumpkins thousnds of feet. INVESTIGTION 3 hunker Pumpkin P 1,500 ft J To mesure the length of throw, two judges nd spotter re involved. One judge is loted t point where the hunker releses the pumpkin. The seond judge is on the right edge of the plying field t point J, distne of 1,500 feet from point. fter throw, the spotter stnds t the lotion P t whih the pumpkin lnded. The two judges use trnsits to mesure nd J, respetively. This informtion is then used to derive the length P of the throw.. In 2004, Old Glory set new world Punkin hunkin reord in the dult ir nnons tegory. To mesure the reord throw, the judges mesured t 30.0 nd J t Wht ws the distne P of the reord throw? LESSON 2 Using Trigonometry in ny Tringle 505

5 On Your Own FOR FULL REDIT, SET-UP/WORK MUST E SHOWN ON LL PROLEMS /12. nother world reord ws set in 2004, this y Hypertension in the dult tpult tegory. ssume for this reord throw of 2,112 feet tht m ws i. Wht ws the distne JP from the position of the judge t J to the position of the spotter t P where the pumpkin lnded? ii. Wht ws the mesure of J?. Suppose in pumpkin throw, m = 33.8 nd the distne JP is 1,435 feet. Explin why this is not enough informtion to uniquely determine the length P of the throw. 8 Suppose in,,, nd re the lengths of the sides opposites ngles,, nd, respetively. Find the mesures of the three unknown prts of to the nerest tenth, given eh of the following sets of onditions.. = 5.4 m, m = 58.0, m = = 6.9 m, m = 47.2, m = = 12.9 m, = 16.3 m, = 8.8 m d. = 9.7 m, = 10.2 m, m = 67.4 onnetions 9 To use the usul formul to find the re of tringle, you need to know the length of one side nd the ltitude to tht se. If you re given the lengths of two sides nd the mesure of the inluded ngle, you n use the Lw of Sines to find the re of the tringle.. Use the three digrms elow to show tht in every se, re of = _ 1 sin. 2 se I se II se III h h. Explin s preisely s you n why it is lso true tht re of = _ 1 2 sin = _ 1 sin. 2. Determine the res of tringles with the following side lengths nd ngle mesures: i. m = 73, = 12, = 17 ii. m = 118, = 21, = UNIT 7 Trigonometri Methods