Yellowknife km Vancouver km NEL


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1 ic tio n Yellowknife Pr e Pu bl 1566 km 67.3 Vncouver km
2 hpter 3 tio n cute Tringle Trigonometry ic LERNING GOLS You will be ble to develop your sptil sense by Pr e Pu bl? Using the sine lw to determine side lengths nd ngle mesures in cute tringles Using the cosine lw to determine side lengths nd ngle mesures in cute tringles Solving problems tht cn be modelled using cute tringles? Who would need to know n ccurte distnce from Yellowknife to Winnipeg? Why cn you not use sine, cosine, or tngent rtios directly to determine the distnce? Winnipeg 113
3 YOU WILL NEED clcultor Field lcrosse, nd s ntionl sport, originted with First Ntions peoples, probbly in centrl North meric. Hundreds of yers go, both the number of plyers nd the size of the field were much greter thn in the modern gme. The Iroquois peoples of wht is now southern Ontrio nd western New York my hve been the first to limit the number of plyers to 12 or Getting Strted Lcrosse Trigonometry Dniel is bout to tke shot t field lcrosse net. He estimtes his current position, s shown below.? 5' 6' Dniel How cn you use these estimtes to determine the width of the net?. Does Dniel s position form right tringle with the golposts?. primry trigonometric rtio cnnot be used to determine the width of the net directly. Explin why.. opy the tringle tht includes Dniel s position in the digrm bove. dd line segment so tht you cn determine height of the tringle using trigonometry. D. Determine the height of the tringle using primry trigonometric rtio. E. rete pln tht will llow you to determine the width of the lcrosse net using the two right tringles you creted. F. rry out your pln to determine the width of the net. 65 gol 114 hpter 3 cute Tringle Trigonometry
4 WHT DO You Think? Decide whether you gree or disgree with ech sttement. Explin your decision. 1. When using proportionl resoning, if you pply the sme opertion, using the sme number, to ll the terms, the rtios remin equivlent. 2. If you know the mesures of two ngles nd the length of ny side in n cute tringle, you cn determine ll the other mesurements. This tem from ughnwg toured Englnd in 1867, demon strting the gme. s result of the tour, lcrosse clubs were estblished in Englnd. The gme then spred round the world. The ndin tem won the bronze medl t the 2009 Women s Lcrosse World up in Prgue, zech Republic. Getting Strted 115
5 3.1 Exploring Side ngle Reltionships in cute Tringles YOU WILL NEED dynmic geometry softwre OR ruler nd protrctor Inukshuks cn hve mny menings. Some inukshuks direct trvel, some indicte rich fishing or hunting res, nd some wrn of dnger. GOL Explore the reltionship between ech side in n cute tringle nd the sine of its opposite ngle. EXPLORE the Mth s they explore the North, the Inuit leve stone cirns, clled inukshuks, s mrkers for those who follow in their pth. You hve used the primry trigonometric rtios to determine side lengths nd ngle mesures in right tringles. n you use primry trigonometric rtios to determine unknown sides nd ngles in ll cute tringles? hoose one of the tringles below. The first tringle is scle digrm of the side of the inukshuk shown. The second tringle represents generl cute tringle cm 46.5? 50.4 h 39.7 cm 37.3 cm 83.3 Wht re two equivlent expressions tht represent the height of ^? Reflecting. Find clssmte who chose different tringle thn you did. ompre ech set of expressions. How re they the sme nd how re they different?. If you drew the height of ^ from different vertex, how would the expressions for tht height be different? Explin.. rete n eqution using the expressions you creted in prt. Show how your eqution cn be written so tht ech rtio in the eqution involves side nd n ngle. D Repet for the expressions you 80 described in prt. 28 cm D. Explin how you could determine the mesure of /E in F this cute tringle.? E 37 cm c h b 116 hpter 3 cute Tringle Trigonometry
6 In Summry Key Ide length of opposite side The rtios of re equivlent for ll three sin 1ngle2 side ngle pirs in n cute tringle. Need to Know In n cute tringle, ^, sin 5 b sin 5 c sin FURTHER Your Understnding 1. For ech cute tringle, i) copy the tringle nd lbel the sides. ii) write two expressions for the height of ech tringle, nd use your expressions to crete equivlent rtios. ) L b) X N Y M Z 2. i) Sketch tringle tht corresponds to ech eqution below. ii) Solve for the unknown side length or ngle mesure. Round your nswer to one deciml plce. w ) sin c) sin 60 sin M sin 72 k b) sin d) sin 85 sin Y sin Michel clims tht if x nd y re sides in n cute tringle, then: x sin Y 5 y sin X Do you gree or disgree? Justify your decision. 4. If you wnt to determine n unknown side length or ngle mesure in n cute tringle, wht is the minimum informtion tht you must hve? opposite side 5. Do you think the rtios of re equivlent for ll three sin 1ngle2 side ngle pirs in right tringle? onstruct two right tringles, nd mesure their sides nd ngles. Use your mesurements to test your conjecture. b c ommuniction The expression x sin Y is product. It is equivlent to x(sin Y ). 3.1 Exploring Side ngle Reltionships in cute Tringles 117 Tip
7 3.2 Proving nd pplying the Sine Lw YOU WILL NEED ruler protrctor clcultor EXPLORE The ngles in n cute tringle mesure 40, 55, nd 85. ould two of the side lengths be 5 cm nd 4 cm? Explin. Explin the steps used to prove the sine lw. Use the lw to solve tringles. INVESTIGTE the Mth In Lesson 3.1, you discovered side ngle reltionship in cute tringles. efore this reltionship cn be used to solve problems, it must be proven to work in ll cute tringles. onsider en s proof: In ^ D, sin 5 opposite hypotenuse sin 5 D c c sin 5 D GOL c D c sin 5 b sin c sin sin 5 b c sin 5 b sin b In ^ D, sin 5 opposite hypotenuse sin 5 D b b sin 5 D Step 1 I drew n cute tringle with height D. Step 2 I wrote equtions for the sine of / nd the sine of / in the two right tringles. Step 3 I set the expressions for D equl to ech other. 118 hpter 3 cute Tringle Trigonometry
8 c In ^ E, sin 5 E c c sin 5 E c sin 5 sin c 5 sin sin c sin 5 sin sin 5 b sin 5 Reflecting E In ^ E, sin 5 E sin 5 E c sin Step 4 I hd expressions tht involved sides b nd c nd / nd /, but I lso needed n expression tht involved nd /. I drew height from to nd developed two expressions for E. Step 5 I set the expressions for E equl to ech other. Step 6 I set ll three rtios equl to ech other.? How cn you improve en s explntion of his proof?. Work with prtner to explin why en drew height D in step 1.. In step 2, he creted two different expressions tht involved D. Explin why.. Explin why he ws ble to set the expressions for D equl in step 3. D. Explin wht en did to rewrite the eqution in step 3. E. In steps 4 nd 5, en drew different height E nd repeted steps 2 nd 3 for the right tringles this creted. Explin why. F. Explin why he ws ble to equte ll three rtios in step 6 to crete the sine lw. G. Why did en not use the cosine rtio or tngent rtio to describe the heights of his cute tringle? H. If en drew perpendiculr line segment from vertex to side, which pir of rtios in the sine lw do you think he could show to be equl? I. Why does it mke sense tht the sine lw cn lso be written in the form sin 5 sin 5 sin? b c sine lw In ny cute tringle, sin 5 b sin 5 c sin 3.2 Proving nd pplying the Sine Lw 119 b c
9 PPLY the Mth exmple 1 Using resoning to determine the length of side tringle hs ngles mesuring 80 nd 55. The side opposite the 80 ngle is 12.0 m in length. Determine the length of the side opposite the 55 ngle to the nerest tenth of metre. Elizbeth s Solution c m sin 5 b sin 12.0 sin 80 5 b sin 55 sin b b 5 sin 55 sin 80 sin 55 b sin sin 80 b 5 b b The length of is 10.0 m. b I could use the sine lw if I knew n opposite side ngle pir, plus one more side or ngle in the tringle. I knew nd / nd I wnted to know b, so I relted, b, sin, nd sin using sin 5 b. Since b ws in the numertor, sin I could multiply both sides by sin 55 to solve for b. Your Turn Using ^ bove, determine the length of to the nerest tenth of metre. I nmed the tringle nd decided tht the 80 ngle ws /. Then I sketched the tringle, including ll of the informtion vilble. I knew tht the third ngle, /, hd to mesure 45, becuse the ngles of tringle dd to 180. I needed to determine b. Since the tringle does not contin right ngle, I couldn t use the primry trigonometric rtios. I rounded to the nerest tenth. It mde sense tht the length of is shorter thn the length of, since the mesure of / is less thn the mesure of /. 120 hpter 3 cute Tringle Trigonometry
10 exmple 2 Toby uses chins ttched to hooks on the ceiling nd winch to lift engines t his fther s grge. The chins, the winch, nd the ceiling re rrnged s shown. Toby solved the tringle using the sine lw to determine the ngle tht ech chin mkes with the ceiling to the nerest degree. He clims tht u 540 nd 554. Is he correct? Explin, nd mke ny necessry corrections. Snjy s Solution I know Toby s clcultions re incorrect, since must be the smllest ngle in the tringle. sin sin sin 86 b 5 1.9sin b sin 86 sin b Solving problem using the sine lw sin sin u u Toby ws incorrect. The correct mesures of the ngles re: 8 43 nd u m 2.8 m ceiling m In ny tringle, the shortest side is cross from the smllest ngle. Since 1.9 m is the shortest side,,u. Toby s vlues do not meet this condition. To correct the error, I used the sine lw to determine. I multiplied both sides by 1.9 to solve for sin. Then I evluted the right side of the eqution. I used the fct tht ngles in tringle dd to 180 to determine u. My determintions re resonble, becuse the shortest side is opposite the smllest ngle. Tip ommuniction Greek letters re often used s vribles to represent the mesures of unknown ngles. The most commonly used letters re u (thet), (lph), b (bet), nd g (gmm). 3.2 Proving nd pplying the Sine Lw 121
11 ommuniction Tip Directions re often stted in terms of north nd south on compss. For exmple, N30 E mens trvelling in direction 30 est of north. S45 W mens trvelling in direction 45 west of south. N N30 E W S45 W 45 S 30 E exmple 3 Using resoning to determine the mesure of n ngle The cptin of smll bot is delivering supplies to two lighthouses, s shown. His compss indictes tht the lighthouse to his left is locted t N30 W nd the lighthouse to his right is locted t N50 E. Determine the compss direction he must follow when he leves lighthouse for lighthouse. W nthony s Solution 9 km km? sin 5 sin 9 km N S 12 km E I drew digrm. I lbelled the sides of the tringle I knew nd the ngle I wnted to determine. I knew,, nd /, nd I wnted to determine /. So I used the sine lw tht includes these four quntities. I used the proportion with sin nd sin in the numertors so the unknown would be in the numertor. 122 hpter 3 cute Tringle Trigonometry
12 sin 9 5 sin sin 80 b 5 9sin 9 12 b sin 80 sin b sin / 5 sin / The mesure of / is km W The cptin must hed N82 W from lighthouse. Your Turn N 12 km S 48 E 40 I substituted the given informtion nd then solved for sin. The nswer seems resonble. / must be less thn 80, becuse 9 km is less thn 12 km. I drew digrm nd mrked the ngles I knew. I knew estwest lines re ll prllel, so the lternte interior ngle t must be 40. The line segment from lighthouse to lighthouse mkes n 8 ngle with westest. I subtrcted this from 90 to determine the direction west of north. In ^ bove, is bout 9.6 km. Use the sine lw to determine /. Verify your nswer by determining the sum of the ngles. The Fisgrd Lighthouse in Victori, ritish olumbi, ws the first lighthouse built on nd s west cost nd is still in opertion tody. 3.2 Proving nd pplying the Sine Lw 123
13 In Summry Key Ide The sine lw cn be used to determine unknown side lengths or ngle mesures in cute tringles. Need to Know You cn use the sine lw to solve problem modelled by n cute tringle when you know: two sides nd the ngle opposite known side. two ngles nd ny side. or If you know the mesures of two ngles in tringle, you cn determine the third ngle becuse the ngles must dd to 180. When determining side lengths, it is more convenient to use: sin 5 b sin 5 c sin When determining ngles, it is more convenient to use: sin 5 sin b 5 sin c HEK Your Understnding 1. Write three equivlent rtios using the sides nd ngles in the tringle t the right. 2. ) Determine length b to the nerest tenth of centimetre. Q s b) Determine the mesure of u to the nerest degree cm 72 b cm 44 r 29.5 cm S q R 124 hpter 3 cute Tringle Trigonometry
14 PRTISING 3. Determine the indicted side lengths to the nerest tenth of unit nd the indicted ngle mesures to the nerest degree. ) D d) N L cm 22.5 cm 45.2 cm 68 F d E b) 40.0 cm e) b c) X f ) 3.0 cm 88 y 25 Y Z 4. Scott is studying the effects of environmentl chnges on fish popultions in his summer job. s prt of his reserch, he needs to know the distnce between two points on 74 Lke Lberge, Yukon. Scott mkes Lke the mesurements shown nd uses Lberge the sine lw to determine the lke s length s 36.0 km km ) gthe, Scott s reserch prtner, sys tht his nswer is incorrect. 52 Explin how she knows. b) Determine the distnce between the two points to the nerest tenth of kilometre. 5. n rchitect designed house nd must give more instructions to the builders. The rfters tht hold up the roof re equl in length. The rfters extend beyond the supporting wll s shown. How long re the rfters? Express your nswer to the nerest inch. 50 Q 2.9 m J 80 K 3.2 Proving nd pplying the Sine Lw 125 S M 5.0 cm 6.5 cm R j 6.7 m 70 36' L 1' The cidity of northern lkes my be ffected by cid rin nd snow cused by development.
15 Gimli, Mnitob, is fishing villge with rich Icelndic heritge. 6. Drw lbelled digrm for ech tringle. Then determine the required side length or ngle mesure. ) In ^SUN, n 5 58 cm, /N 5 38, nd /U Determine the length of side u to the nerest centimetre. b) In ^PQR, /R 5 73, /Q 5 32, nd r 5 23 cm. Determine the length of side q to the nerest centimetre. c) In ^TM, t 5 8 cm, m 5 6 cm, nd /T Determine the mesure of /M to the nerest degree. d) In ^WXY, w cm, y cm, nd /W Determine the mesure of /Y to the nerest degree. 7. In ^T, / 5 32, /T 5 81, nd c m. Solve the tringle. Round sides to the nerest tenth of metre. 8. ) Determine the vlue of n to the L nerest tenth using i) primry trigonometric rtio. 10 cm ii) the sine lw. n b) Explin why your nswers for prt ) re the sme. Do others in your clss 36.9 gree with your explntion? M N 9. Jnice is siling from Gimli on Lke Winnipeg to Grnd ech. She hd plnned to sil 26.0 km in the direction S71 E; however, the wind nd current pushed her off course. fter severl hours, she discovered tht she hd ctully been siling S79 E. She checked her mp nd sw tht she must sil S18 W to rech Grnd ech. Determine, to the nerest tenth of kilometre, the distnce remining to Grnd ech. ) Drw digrm of this sitution, then compre it with clssmte s. Mke ny djustments needed in your digrms. b) Solve the problem. 10. telephone pole is supported by two wires on opposite sides. t the top of the pole, the wires form n ngle of 60. On the ground, the ends of the wires re 15.0 m prt. One wire mkes 45 ngle with the ground. ) Drw digrm of this sitution, then compre it with clssmte s. b) How long re the wires, nd how tll is the pole? Express your nswers to the nerest tenth of metre. 11. In ^PQR, /Q 5 90, r 5 6, nd p 5 8. Explin two different wys to determine the mesure of /P. Shre your explntion with prtner. How might you improve your explntion? 12. Stell decided to ski to friend s cbin. She skied 10.0 km in the direction N40 E. She rested, then skied S45 E nd rrived t the cbin. The cbin is 14.5 km from her home, s the crow flies. Determine, to the nerest tenth of kilometre, the distnce she trvelled on the second leg of her trip. 126 hpter 3 cute Tringle Trigonometry
16 13. bridge hs been built cross gorge. Jordn wnts to bungee jump from the bridge. One of the things she must know, to mke the jump sfely, is the depth of the gorge. She mesured the gorge s shown. Determine the depth of the gorge to the nerest tenth of metre. 14. Sketch n cute tringle. ) List three pieces of informtion bout the tringle s sides nd ngles tht would llow you to solve for ll the other side lengths nd ngle mesures of the tringle. b) List three pieces of informtion bout the tringle s sides nd ngles tht would not llow you to solve the tringle. 15. Jim sys tht the sine lw cnnot be used to determine the length of side c in ^. Do you gree or disgree? Explin. losing 16. Suppose tht you know the length of side p in ^ PQR, s well s the mesures of /P nd /Q. Wht other sides nd ngles could you determine? Explin to clssmte how you would determine these mesurements. Extending 17. In ^, / 5 58, / 5 74, nd b 5 6. Determine the re of ^ to one deciml plce. 18. n isosceles tringle hs two sides tht re 10.0 cm long nd two ngles tht mesure 50. line segment bisects one of the 50 ngles nd ends t the opposite side. Determine the length of the line segment to the nerest tenth of centimetre. 19. Use the sine lw to write rtio tht is equivlent to ech expression for ^. sin sin ) b) c) sin c c sin 3.2 m m 70 c 3.0 m Proving nd pplying the Sine Lw 127
17 Study id See Lesson 3.1. Try Midhpter Review Questions 1 to 3. Study id See Lesson 3.2, Exmples 1 to 3. Try Midhpter Review Questions 4 to 9. 3 Midhpter Review FREQUENTLY SKED Questions Q: Wht is the sine lw, nd wht is it used for? : The sine lw describes the reltionship between the sides nd their opposite ngles in tringle. In ^, sin 5 b sin 5 c b sin or c sin 5 sin 5 sin b c The sine lw cn be used to determine unknown side lengths nd ngle mesures in cute nd right tringles. Q: When cn you use the sine lw? : Use the sine lw if you know ny three of these four mesurements: two sides nd their opposite ngles. The sine lw llows you to determine the unknown length or ngle mesure. If you know ny two ngles in tringle, you cn determine the third ngle. For exmple, the sine lw cn be used to determine the length of in the tringle t the right. Side is opposite /, which mesures 72. Side is of length 500 m nd is opposite / m / / 5 56 Use the sine lw to write n eqution nd solve for. sin sin 56 5 sin sin 56 b The length of is bout 574 m. 128 hpter 3 cute Tringle Trigonometry
18 PRTISING L b) Lesson Write the equivlent sine lw rtios for cute 8 cm tringle XYZ. M 2. ) Sketch n cute tringle tht illustrtes the f d 5 sin F sin D Do you gree or disgree? Explin. Lesson Determine the vlues indicted with vribles to the nerest tenth of unit. ) 4.0 cm 50 Pu b) x y e x cm 5. Solve ech tringle. Where necessry, round nswers to the nerest tenth of unit. ) Pr dditionl informtion do you need to know if you wnt to use the sine lw to solve the tringle? 7. Two Jsper Ntionl Prk rngers in their fire towers spot fire km ) Which tower is closer to the fire? Explin. b) Determine the distnce, to the nerest tenth of kilometre, from this tower to the fire. bl 4.5 cm N 6. In ^ XYZ, the vlues of x nd z re known. Wht ic 3. ^DEF is n cute tringle. Nzir clims tht 14 tio n reltionship described in the eqution below. x 10 5 sin 60 sin 80 b) Determine the vlue of x to the nerest tenth s hloe nd Ivn re pddling north on Lc L Ronge in Ssktchewn, they notice cmpsite hed, t N52 W. They continue pddling north for 800 m, which tkes them pst the cmpsite. The cmpsite is then t S40 W. How fr wy, to the nerest metre, is the cmpsite from their position t the second sighting? 9. Determine, to the nerest tenth of centimetre, the perimeter of ech isosceles tringle. O ) b) D cm 70 E 25 cm F P cm Midhpter Review Q
19 3.3 Proving nd pplying the osine Lw YOU WILL NEED ruler protrctor clcultor EXPLORE One side of right tringle is 8 cm. One ngle is 50. Wht could the other side lengths be? cosine lw In ny cute tringle, 2 5 b 2 1 c 2 2 2bc cos b c 2 2 2c cos c b 2 2 2b cos b c GOL Explin the steps used to prove the cosine lw. Use the cosine lw to solve tringles. INVESTIGTE the Mth The sine lw cnnot lwys help you determine unknown ngle mesures or side lengths. onsider these tringles: s 3.2 m e 2.6 m D R Q 66 q? F? f 3.6 m r 3.1 m d 2.5 m E S where 3.1 sin R sin S 5 q sin 66 where sin E sin D sin F 3.6 There re two unknowns in ech pir of equivlent rtios, so the pirs cnnot be used to solve for the unknowns. nother reltionship is needed. This reltionship is clled the cosine lw, nd it is derived from the Pythgoren theorem. efore this reltionship cn be used to solve problems, it must be proven to work in ll cute tringles. onsider Hether s proof of the cosine lw: Step 1 c I drew n cute tringle. b h Then I drew the height from x to nd lbelled the D y intersection point s point D. I lbelled this line segment h. I lbelled D s x nd D s y. 130 hpter 3 cute Tringle Trigonometry
20 h 2 5 c 2 2 x 2 h 2 5 b 2 2 y 2 Step 2 I wrote two different expressions for h 2.? c 2 2 x 2 5 b 2 2 y 2 c 2 5 x 2 1 b 2 2 y 2 x 5 2 y, so c y2 2 1 b 2 2 y 2 c y 1 y 2 1 b 2 2 y 2 c b 2 2 2y cos 5 y b, so b cos 5 y c b 2 2 2y c b 2 2 2b cos How cn you improve Hether s explntions in her proof of the cosine lw?. Work with prtner to explin why she drew height D in step 1.. In step 2, Hether creted two different expressions tht involved h 2. Explin how she did this.. Explin why she ws ble to set the expressions for h 2 equl in step 3. D. In step 4, Hether eliminted the vrible x. Explin how nd why. E. Explin how she determined n equivlent expression for y in step 5. F. Explin why the finl eqution in step 6 is the most useful form of the cosine lw. Reflecting Step 3 I set the two expressions equl to ech other nd solved for c 2. Step 4 I wrote n equivlent eqution tht only used the vrible y nd simplified. Step 5 I determined n equivlent expression for y. Step 6 I substituted the expression b cos for y in my eqution. G. Les wrote similr proof, but he substituted 2 x for y insted of 2 y for x in the eqution in step 3. How would his result differ from Hether s? H. Frnçois strted his proof by drwing height from to. How would this ffect his finl result? I. i) Explin why you cn use the cosine lw to determine the unknown side q in ^QRS on the previous pge. ii) Explin why you cn use the cosine lw to determine the unknown /F in ^DEF on the previous pge. 3.3 Proving nd pplying the osine Lw 131
21 PPLY the Mth exmple 1 Using resoning to determine the length of side Determine the length of to the nerest metre. Justin s Solution b 32 m 58? c 40 m 2 5 b 2 1 c 2 2 2bc cos (32)(40) cos cos cos " is 36 m. Your Turn 32 m m fter determining the length of in ^ bove, Justin used the sine lw to determine tht the mesure of / is 50, then concluded tht the mesure of / must be 72. Use the cosine lw to verify his solution for I lbelled the sides with letters. I couldn t use the sine lw, becuse I didn t know side length nd the mesure of its opposite ngle. I knew the lengths of two sides (b nd c) nd the mesure of the contined ngle between these sides (/ ). I hd to determine side, which is opposite /. I chose the form of the cosine lw tht includes these four vlues. Then I substituted the vlues I knew into the cosine lw. ^. 132 hpter 3 cute Tringle Trigonometry
22 exmple 2 Using resoning to determine the mesure of n ngle The digrm t the right shows the pln for roof, with support bem DE prllel to. The locl building code requires the ngle formed t the pek of roof to fll within rnge of 70 to 80 so tht snow nd ice will not build up. Will this pln pss the locl building code? Lunne s Solution: Substituting into the cosine lw, then rerrnging 10'? 19.5' 20' , b 5 20, nd c 5 10 b c 2 2 2c cos cos (cos ) (cos ) cos cos b 5 / / The ngle formed t the pek of the roof is 78. This pln will pss the locl building code. 10' D E 20' 19'6" I drew sketch, removing the support bem since it isn t needed to solve this problem. The pek of the roof is represented by /. I lbelled the sides I knew in the tringle. I wrote ll the lengths using the sme unit, feet. Since I only knew the lengths of the sides in the tringle, I couldn t use the sine lw. I hd to determine /, so I decided to use the form of the cosine lw tht contined /. I substituted the side lengths into the formul nd simplified. I hd to isolte cos before I could determine /. My nswer is resonble becuse / should be the ngle with the lrgest mesure in the tringle. 78 lies within the cceptble rnge of 70 to Proving nd pplying the osine Lw 133
23 Emilie s Solution: Rerrnging the cosine lw before substituting? c 10' b 20' 19.5' b c 2 2 2c cos I drew digrm, lbelling the sides nd ngles. I wrote ll the side lengths in terms of feet. b 2 1 2c cos c 2 2 2c cos 1 2c cos I decided to rerrnge the formul to b 2 1 2c cos c 2 solve for cos by dding 2c cos to b 2 1 2c cos 2 b c 2 2 b 2 both sides of the eqution. Then I 2c cos c 2 2 b 2 subtrcted b 2 from both sides. Finlly 2c cos c 2 2 b 2 I divided both sides by 2c. 2c 2c cos c 2 2 b 2 2c I substituted the informtion tht I knew into the cos rerrnged formul nd evluted the right side. cos cos / 5 cos / The ngle formed t the pek of the roof is 78. This pln psses the locl building code. Your Turn ) ompre Lunne s Solution nd Emilie s Solution. Wht re the dvntges of ech strtegy? b) Which strtegy do you prefer for this problem? Explin. c) Use your strtegy nd the cosine lw to determine / in ^ bove. Since I wnted to determine / nd I knew the length of ll three sides, I wrote the form of the cosine lw tht contins /. I rounded to the nerest degree. The vlue of this ngle is within the cceptble rnge. 134 hpter 3 cute Tringle Trigonometry
24 exmple 3 Solving problem using the cosine lw threepointed str is mde up of n equilterl tringle nd three congruent isosceles tringles. Determine the length of ech side of the equilterl tringle in this threepointed str. Round the length to the nerest centimetre. 60 cm cm Dkod s Solution X 60 cm cm Y Z 1Y Z X Y X Z X Y 21X Z 2 cos 1/Y X Z 2 1Y Z cos 20 1Y Z Y Z YZ 5 " YZ Ech side of the equilterl tringle hs length of 21 cm. The world s lrgest Ukrinin Ester egg (clled pysnk) is locted in Vegreville, lbert. It is decorted with 2208 equilterl tringles nd 524 threepointed strs. I nmed the vertices of one of the isosceles tringles. I knew two sides nd the contined ngle in ech isosceles tringle, so I used the cosine lw to write n eqution tht involved YZ. Then I substituted the informtion tht I knew. 3.3 Proving nd pplying the osine Lw 135
25 In Summry Key Ide The cosine lw cn be used to determine n unknown side length or ngle mesure in n cute tringle. Need to Know b c 2 5 b 2 1 c 2 2 2bc cos b c 2 2 2c cos c b 2 2 2b cos You cn use the cosine lw to solve problem tht cn be modelled by n cute tringle when you know: two sides nd the ll three sides. contined ngle. The contined ngle is the ngle between two known sides. When using the cosine lw to determine n ngle, you cn: substitute the known vlues first, then solve for the unknown ngle. rerrnge the formul to solve for the cosine of the unknown ngle, then substitute nd evlute. HEK Your Understnding 1. Suppose tht you re given ech set of dt for ^. n you use the cosine lw to determine c? Explin. ) 5 5 cm, / 5 52, / 5 43 b) 5 7 cm, b 5 5 cm, / 5 43 c b 136 hpter 3 cute Tringle Trigonometry
26 2. Determine the length of side x to the nerest centimetre. X cm Y 15 cm x 3. Determine the mesure of /P to the nerest degree. PRTISING 4. Determine ech unknown side length to the nerest tenth of centimetre. ) b) D 10.5 cm cm 13.0 cm 11.0 cm 5. Determine the mesure of ech indicted ngle to the nerest degree. ) P b) X 3.5 cm 3.9 cm Q 2.2 cm R Y cm 2.6 cm E 2.2 cm Z 6. Sketch ech tringle, bsed on the given eqution. Then solve for the unknown side length or ngle mesure. Round ll nswers to the nerest tenth of unit. ) w cos 75 b) k cos 50 c) (46)(45) cos Y d) (17)(15) cos G Q 2.3 m P 5.9 m 6.2 m Z F R 3.3 Proving nd pplying the osine Lw 137
27 7. Solve ech tringle. Round ll nswers to the nerest tenth of unit. ) In ^DEF, d cm, e cm, nd /F b) In ^PQR, p m, q m, nd /R c) In ^LMN, l cm, m cm, nd n cm. d) In ^ XYZ, x mm, y mm, nd z mm. 8. The pendulum of grndfther clock is cm long. When the pendulum swings from one side to the other side, the horizontl distnce it trvels is 9.6 cm. ) Drw digrm of the sitution. b) Determine the ngle through which the pendulum swings. Round your nswer to the nerest tenth of degree. 9. Determine the perimeter of ^SRT, if /S 5 60, r 5 15 cm, nd t 5 20 cm. Round your nswer to the nerest tenth of centimetre. 10. prllelogrm hs sides tht re 8 cm nd 15 cm long. One of the ngles in the prllelogrm mesures 70. Explin how you could determine the length of the shorter digonl. 11. ) clock hs minute hnd tht is 20 cm long nd n hour hnd tht is 12 cm long. Determine the distnce between the tips of the hnds t i) 2:00. ii) 10:00. b) Discuss your results for prt ). 12. Emilie mkes stined glss windows to sell t the Festivl du ois in Millrdville, ritish olumbi. Ech piece of glss is surrounded by led edging. Emilie clims tht she cn crete n cute tringle in prt of window using pieces of led tht re 15 cm, 36 cm, nd 60 cm. Is she correct? Justify your decision. The Festivl du ois, one of ritish olumbi s gretest celebrtions of Frenchndin culture, is held in Mrch. 138 hpter 3 cute Tringle Trigonometry
28 13. Two drivers leve their school t the sme time nd trvel on stright rods tht diverge by 70. One driver trvels t n verge speed of 33.0 km/h. The other driver trvels t n verge speed of 45.0 km/h. How fr prt will the two drivers be fter 45 min, to the nerest tenth of kilometre? losing 14. Use the tringle t the right to crete problem tht involves side lengths nd interior ngles. Then describe how to determine the length of side d. Exchnge your problem with clssmte. Extending 15. The distnce from the centre, O, of regulr decgon to ech vertex is 12 cm. Determine the re of the decgon. Round your nswer to the nerest squre centimetre. 16. The centre, O, of regulr pentgon is perpendiculr distnce of 1.5 cm from ech side. Determine the perimeter, to the nerest tenth of centimetre, nd re, to the nerest tenth of squre centimetre, of the pentgon. 17. n ulu is n Inuit llpurpose knife, trditionlly used by women. The metl blde of one type of ulu is roughly tringulr in shpe, with the cutting edge opposite the vertex where the hndle is ttched. The other sides of the ulu re roughly equl. Describe functionl ulu tht hs 14 cm blde, mesured point to point. Include the vertex ngle t the hndle nd the side lengths in your description. 3.3 Proving nd pplying the osine Lw m cm? 35 m O 70 d
29 3.4 Solving Problems Using cute Tringles YOU WILL NEED ruler clcultor EXPLORE Two plnes leve n irport on different runwys t the sme time. One heds S40 W nd the other heds S60 E. rete problem bout the plnes tht cn be solved only by using the cosine lw. Solve the problem. exmple 1 GOL Solve problems using the primry trigonometric rtios nd the sine nd cosine lws. LERN OUT the Mth onnecting n cute tringle model to sitution Determine the ngles of depression, to the nerest degree, for ech cmer. Vld s Solution: Using primry trigonometric rtios nd the cosine lw 6.0 m 4.8 m D 4.5 m 1.5 m 12 m 6.0 m Two security cmers in museum must be djusted to monitor new disply of fossils. The cmers re mounted 6 m bove the floor, directly cross from ech other on opposite wlls. The wlls re 12 m prt. The fossils re displyed in cses mde of wood nd glss. The top of the disply is 1.5 m bove the floor. The distnce from the cmer on the left to the centre of the top of the disply is 4.8 m. oth cmers must im t the centre of the top of the disply.? Wht is the ngle of depression for ech cmer? I drew digrm. I plced the cmers 6 m from the floor, 12 m wy from ech other on opposite wlls t points nd. I wsn t sure where to plce the disply or its centre, D. The disply hd to be closer to cmer since the distnce from cmer to the disply ws only 4.8 m. Subtrcting the disply height from 6 m gve me the distnce from the disply to the horizontl between the cmers. I lbelled the ngles of depression using u nd. 140 hpter 3 cute Tringle Trigonometry
30 sin u u5sin b u m 69.6 D 12 m z z cos z z 5 " z m opposite D 11.3 m hypotenuse sin sin b To monitor the disply effectively, cmer must be djusted to n ngle of depression of 70 nd cmer must be djusted to n ngle of depression of 24. Michel s Solution: Using only primry trigonometric rtios 6.0 m 4.8 m 4.5 m 1.5 m 12 m 6.0 m In the right tringle contining ngle u, the side of length 4.5 m is opposite ngle u, nd the side of length 4.8 m is the hypotenuse. I could use the sine rtio to determine the mesure of u. I knew the lengths of two sides in ^ D nd the ngle between them. So, I ws ble to use the cosine lw to determine the length of the side opposite cmer, which I lbelled z. sed on the sides I knew in the right tringle contining ngle, I wrote n eqution using the sine rtio. E D I drew digrm by plcing the cmers t points nd nd the centre of the disply t point D. I knew point D hd to be closer to cmer becuse the distnce between it nd cmer hd to be greter thn 4.8 m. Subtrcting the disply height from the cmer height gve me the length of DE, the height of ^ D. I lbelled the ngles of depression u nd. 3.4 Solving Problems Using cute Tringles 141
31 sin u u5sin b u m E D 4.5 m 12 m cos E cos E E E E E 4.5 m opposite D 10.3 m djcent tn tn b mer must be djusted to n ngle of depression of 70 nd cmer must be djusted to n ngle of depression of 24 to ensure tht they both point to the centre of the disply. In right tringle DE, DE is opposite ngle u nd D is the hypotenuse. Since I knew the lengths of both sides, I used the sine rtio to determine ngle u. In right tringle DE, E is djcent to ngle u nd D is the hypotenuse. Since I knew the length of D nd the mesure of ngle u, I used the cosine rtio to determine the length of E. To determine the length of E, I subtrcted the length of E from 12. sed on the sides I knew in right tringle DE, I wrote n eqution using the tngent rtio to determine ngle. 142 hpter 3 cute Tringle Trigonometry
32 Reflecting. Why do you think Vld strted his solution by using the right tringle tht contined ngle u insted of the right tringle tht contined ngle?. ould Vld hve determined the vlue of using the sine lw? Explin.. Which solution do you prefer? Justify your choice. PPLY the Mth exmple 2 onnecting cute tringle models to indirect mesurement The world s tllest freestnding totem pole is locted in econ Hill Prk in Victori, ritish olumbi. It ws crved from single cedr log by noted crver hief Mungo Mrtin of the Kwkiutl (Kwkwk wkw), with tem tht included his son Dvid nd Henry Hunt. It ws erected in While visiting the prk, Mnuel wnted to determine the height of the totem pole, so he drew sketch nd mde some mesurements: ground D m I wlked long the shdow of the totem pole nd counted 42 pces, estimting ech pce ws bout 1 m. I estimted tht the ngle of elevtion of the Sun ws bout 40. I observed tht the shdow rn uphill, nd I estimted tht the ngle the hill mde with the horizontl ws bout 5. How cn Mnuel determine the height of the totem pole to the nerest metre? 5 h 3.4 Solving Problems Using cute Tringles 143
33 Mnuel s Solution h is the height of ^, but it is lso side in cute tringle D. /D /D /D /D 5 45 /D /D 5 85 /D /D 5 50 ground D h sin sin 50 h 42 sin 45 b 5 sin 45 sin 45 sin 50 b h 5 sin sin 50 b h The totem pole is 39 m tll. Your Turn m List some sources of error tht my hve occurred in Mnuel s strtegy tht would ffect the ccurcy of his determintion h I needed to determine the ngles in this tringle to be ble to determine h using the sine lw. I used the two ngles I knew in ^ D to determine the third ngle in this tringle, since the ngles in tringle dd to 180. I subtrcted / D from 180 to determine /D, since these re supplementry ngles. I subtrcted /D from 90 to determine /D, since these re complementry ngles. I used the two ngles I determined in ^D to determine the third ngle in this tringle. I dded ll the informtion I determined bout ^D to my sketch. I used the sine lw to write n eqution tht contined h. Then I solved for h. 144 hpter 3 cute Tringle Trigonometry
34 exmple 3 Solving threedimensionl problem rendn nd Din pln to climb the cliff t Dry Islnd ufflo Jump, lbert. They need to know the height of the climb before they strt. rendn stnds t point, s shown in the digrm. He uses clinometer to determine /, the ngle of elevtion to the top of the cliff. Then he estimtes /D, the ngle between the bse of the cliff, himself, nd Din, who is stnding t point D. Din estimtes /D, the ngle between the bse of the cliff, herself, nd rendn. Din s Solution 50 D m ic bl I knew two ngles nd side length in ^D. efore I could determine, I hd to determine /D. I used the fct tht the sum of ll three interior ngles is 180. I used the sine lw to write n eqution tht involved in ^D. Pu sin I didn t hve enough informtion bout ^ to determine the height,. I needed the length of. is in ^, but it is lso in ^D. In ^ D, /D /D 5 70 D 5 sin D sin 60 5 sin 50 sin 70 tio n Determine the height of the cliff to the nerest metre. 60 b 5 sin 50 b sin 50 sin 70 To solve for, I multiplied both sides of the eqution by sin b sin e 5 sin 50 tn Pr tn tn The height of the cliff is 196 m. I knew tht ^ is right tringle. I lso knew tht is opposite the 76 ngle nd is djcent to it. So, I used the tngent rtio to write n eqution tht involved. b Solving Problems Using cute Tringles 145
35 Your Turn rete threedimensionl problem tht cn be solved using Din s strtegy. Wht fetures of your problem mke it necessry to use two tringles to solve the problem? In Summry Key Ide The sine lw, the cosine lw, the primry trigonometric rtios, nd the sum of ngles in tringle my ll be useful when solving problems tht cn be modelled using cute tringles. Need to Know To decide whether you need to use the sine lw or the cosine lw, consider the informtion given bout the tringle nd the mesurement to be determined. Mesurement to be Informtion Given Determined Use two sides nd the ngle opposite one of the sides ngle sine lw two ngles nd side side sine lw two sides nd the contined ngle side cosine lw three sides ngle cosine lw Drwing clerly lbelled digrm mkes it esier to select strtegy for solving problem. 146 hpter 3 cute Tringle Trigonometry
36 HEK Your Understnding 1. Explin how you would determine the indicted ngle mesure or side length in ech tringle. ) b) c) 2.7 cm c 3.0 cm 2.9 cm cm 1.8 cm 2.0 cm 2. ) Use the strtegies you described to determine the mesurements indicted in question 1. Round your nswers to the nerest tenth of unit. b) ompre your nswers for questions 1 nd 2) with clssmte s nswers. Which strtegy seems to be most efficient for ech? PRTISING 3. kyk leves Rnkin Inlet, Nunvut, nd heds due est for 5.0 km, s shown in the digrm. t the sme time, second kyk trvels in direction S60 E from the inlet for 4.0 km. How fr prt, to the nerest tenth of kilometre, re the kyks? ) Describe how you cn solve the problem. b) Determine the distnce between the kyks. N L 4. How long, to the nerest inch, is ech rfter in the roof shown? rfter 5.0 km km rfter ' 5. crne stnds on top of building, s shown. ) How fr is the point on the ground from the bse of the building, to the nerest tenth of metre? b) How tll is the crne?? E m 3.4 Solving Problems Using cute Tringles 147
37 Ships re sometimes delibertely sunk (scuttled) to form brekwters nd rtificil reefs. 6. tree is growing on hillside, s shown. The hillside is inclined t n ngle of 15 to the horizontl. The tree csts shdow uphill. How tll is the tree, to the nerest metre? ) Describe how you cn solve the problem. b) Determine the height of the tree rdr opertor on ship discovers lrge sunken vessel lying prllel to the ocen surfce, 200 m directly below the ship. The length of the vessel is clue to which wreck hs been found. The rdr opertor mesures the ngles of depression to the front nd bck of the sunken vessel to be 56 nd 62. How long, to the nerest tenth of metre, is the sunken vessel? 8. Fred nd gnes re 520 m prt. s rendn flies overhed in n irplne, they estimte the ngle of elevtion of the irplne. Fred, looking south, estimtes the ngle of elevtion to be 60. gnes, looking north, estimtes it to be 40. Wht is the ltitude of the irplne, to the nerest tenth of metre? 9. Two support wires re fstened to the top of communictions tower from points nd on the ground. The points re on opposite sides of the tower nd in line. One wire is 18 m long, nd the other wire is 12 m long. The ngle of elevtion of the longer wire to the top of the tower is 38. ) How tll is the tower, to the nerest tenth of metre? b) How fr prt re points nd, to the nerest tenth of metre? 57 7 m 10. regulr pentgon is inscribed in circle with centre O, s shown in the digrm. ) Work with prtner to develop strtegy to determine the perimeter of the pentgon. b) rry out your strtegy to determine the perimeter to the nerest tenth of centimetre. O 10 cm 148 hpter 3 cute Tringle Trigonometry
38 11. Ryn is in police helicopter, 400 m directly bove the Se to Sky highwy ner Whistler, ritish olumbi. When he looks north, the ngle of depression to cr ccident is 65. When he looks south, the ngle of depression to the pproching mbulnce is 30. ) How fr wy is the mbulnce from the scene of the ccident, to the nerest tenth of metre? b) The mbulnce is trvelling t 80 km/h. How long will it tke the mbulnce to rech the scene of the ccident? 12. The rdr screen in the irtrffic control tower t the Edmonton Interntionl irport shows tht two irplnes re t the sme ltitude. ccording to the rnge finder, one irplne is 100 km wy, in the direction N60 E. The other irplne is 160 km wy, in the direction S50 E. ) How fr prt re the irplnes, to the nerest tenth of kilometre? b) If the irplnes re pproching the irport t the sme speed, which irplne will rrive first? 13. In prllelogrm, two djcent sides mesure 10 cm nd 12 cm. The shorter digonl is 15 cm. Determine, to the nerest degree, the mesures of ll four ngles in the prllelogrm. 14. Two students decided to determine the ltitude, h, of promotionl blimp flying over McMhon Stdium in lgry. The students mesurements re shown in the digrm. Determine h to the nerest tenth of metre. Explin ech of your steps. D h m Mth in ction How Good Is Your Peripherl Vision? When you stre stright hed, you cn still see objects to either side. This is clled peripherl vision. It cn be mesured using n ngle. For exmple, the ngle for your right eye would be swept out from point directly in front of your nose to the point where you cn no longer see objects on the fr right. This ngle is bout 60 for those with norml peripherl vision. Work with prtner or in smll group. Mke pln to mesure the peripherl vision of your eyes. The only mterils you cn use re pencil, metre stick, nd string. Test your pln. Wht is your peripherl vision? Evlute your pln. Wht djustments did you need to mke during the test? re you stisfied tht your pln worked well? Explin. 3.4 Solving Problems Using cute Tringles 149
39 History onnection Fire Towers nd Lookouts losing 15. rete rellife problem tht cn be modelled by n cute tringle. Exchnge the problem you creted with clssmte. Sketch the sitution in your clssmte s problem, nd explin wht must be done to solve it. Extending E 16. The Gret Pyrmid t Giz in Egypt hs squre bse with sides tht re m long. The D distnce from the top of the F G pyrmid to ech corner of the bse ws originlly m m ) Determine the pex ngle of fce of the pyrmid (for exmple, /E2 to the nerest degree. b) Determine the ngle tht ech fce mkes with the bse (for exmple, /EGF 2 to the nerest degree. 17. ut out two pper strips, ech 5 cm wide. Ly them cross ech other s shown t the right. Determine the re of the overlpping region. Round your nswer to the nerest tenth of squre centimetre m For bout 100 yers, observers hve been wtching for forest fires. Perched in lookouts on high ground or on tll towers (there re three fire towers in lbert tht re 120 m high), the observers wtch for smoke in the surrounding forest. When the observers see signs of fire, they report the sighting to the Fire entre. The first observers my hve hd s little s mp, binoculrs, nd horse to ride to the nerest sttion to mke report. Tody, observers hve instruments clled liddes nd report using rdios. liddes consist of locl mp, fixed in plce, with the tower or lookout t the centre of the mp. ompss directions re mrked in degrees round the edge of the mp. rnge finder rottes on n rc bove the mp. The observer rottes the rnge finder to get the smoke in both sights, notes the direction, nd reports it to the Fire entre. computer t the Fire entre cn use the loctions of the fire towers nd directions from two or more observers to fix the loction of the fire. The ltitude nd longitude re then entered into helicopter s flight GPS system. The helicopter s crew flies to the fire, records more ccurte loction, nd reports on the fire.. How could Fire entre use trigonometry to determine the loction of fire?. How could n observer use trigonometry to estimte the size of fire? 150 hpter 3 cute Tringle Trigonometry
40 pplying ProblemSolving Strtegies nlyzing Trigonometry Puzzle Puzzles do not lwys hve precise solutions. They cnnot lwys be solved purely by deduction, lthough logic helps. The Puzzle. elow re seven similr right tringles. Trce the tringles, nd cut them out.. Use ll seven tringles to form single squre, with no overlpping.. If the hypotenuse of the gretest tringle is 10 units long, wht is the re of the squre? The Strtegy D. Describe the strtegy you used to form the squre. E. Describe the strtegy you used to determine the re of the squre. YOU WILL NEED ruler scissors 3.4 Solving Problems Using cute Tringles 151
41 3 hpter SelfTest 1. Determine the indicted side length or ngle mesure in ech tringle. Round nswers to the nerest tenth of unit. ) b) cm 78 c 4.1 cm 6.0 cm cm 2. In ^ PQR, /P 5 80, /Q 5 48, nd r 5 20 cm. Solve ^ PQR. Where necessry, round nswers to the nerest tenth of unit. 3. The rdr screen of ost Gurd N rescue ship shows tht two bots re in bot 1 the re, s shown in the digrm. How 45 fr prt re the two bots, to the W E? nerest tenth of kilometre? 4. prllelogrm hs djcent sides tht 50 re 11.0 cm nd 15.0 cm long. The ngle between these sides is 50. Determine S bot 2 the length of the shorter digonl to the nerest tenth of centimetre. 5. Points P nd Q lie 240 m prt in line with nd on opposite sides of communictions tower. The ngles of elevtion to the top of the tower from P nd Q re 50 nd 45, respectively. Determine the height of the tower to the nerest tenth of metre. 6. Terry is designing tringulr ptio, s shown. Determine the re of the ptio to the nerest tenth of squre metre. 9.0 m 70 km 100 km 7. In n cute tringle, two sides re 2.4 cm nd 3.6 cm. One of the ngles is 37. How cn you determine the third side in the tringle? Explin. 8. Why do you need both the sine lw nd the cosine lw to determine side lengths in n cute tringle? 7.1 m 8.5 m WHT DO You Think Now? Revisit Wht Do You Think? on pge 115. How hve your nswers nd explntions chnged? 152 hpter 3 cute Tringle Trigonometry
42 FREQUENTLY SKED Questions Q: To use the cosine lw, wht do you need to know bout tringle? : You need to know two sides nd the contined ngle, or three sides in the tringle. For exmple, you cn use the cosine lw to determine the length of p. P 6 cm 72 Q 8 cm p R You know: the lengths of two sides. the mesure of the contined ngle. You cn use the cosine lw to determine the length of the side opposite the contined ngle, p. p cos 72 Solving for p will determine the length. You cn lso use the cosine lw to determine the mesure of /Y. Y 3 hpter Review 4 cm X 5 cm 2 cm You know: the lengths of ll three sides. You cn use the cosine lw to determine the mesure of /Y cos Y cos Y Solving for Y will determine the mesure of /Y. Q: When solving problem tht cn be modelled by n cute tringle, how do you decide whether to use the primry trigonometric rtios, the sine lw, or the cosine lw? : Drw clerly lbelled digrm of the sitution to record wht you know. You my be ble to use primry trigonometric rtio if the digrm involves right tringle. Use the sine lw if you know two sides nd one opposite ngle, or two ngles nd one opposite side. Use the cosine lw if you know ll three sides, or two sides nd the ngle between them. You my need to use more thn one strtegy to solve some problems. Z Study id See Lesson 3.3, Exmples 1 to 3. Try hpter Review Questions 6 to 9. Study id See Lesson 3.4, Exmples 1 to 3. Try hpter Review Questions 10 to 12. hpter Review 153
43 PRTISING Lesson Jne clims tht she cn drw n cute tringle using the following informtion: 5 6 cm, b 5 8 cm, c 5 10 cm, / 5 30, nd / Is she correct? Explin. 2. Which of the following re not correct for cute tringle DEF? d ) sin D 5 f c) f sin E 5 e sin F sin F b) sin E 5 sin D e d Lesson 3.2 d) d sin D 5 sin F f 3. Determine the indicted side or ngle in ech tringle to the nerest tenth of unit. ) b) m 35.4 cm x 31.2 cm 4. Solve ^, if / 5 75, / 5 50, nd the side between these ngles is 8.0 cm. Round nswers to the nerest tenth of unit. 5. llison is flying kite. She hs relesed the entire 150 m bll of kite string. She notices tht the string forms 70 ngle with the ground. Mrc is on the other side of the kite nd sees the kite t n ngle of elevtion of 30. How 150 m fr is Mrc from M llison, to the nerest tenth of metre? Lesson Which of these is not form of the cosine lw for ^? Explin. ) 2 5 b 2 1 c 2 2 2bc cos b) c b 2 2 2b cos c) b c 2 2 2c cos Determine the indicted side or ngle. Round nswers to the nerest tenth of unit. ) b) 7.0 m 5.0 m 8. Solve ^, if / 5 58, b cm, nd c cm. Round nswers to the nerest tenth of unit. 9. Two irplnes leve the Hy River irport in the Northwest Territories t the sme time. One irplne trvels t 355 km/h. The other irplne trvels t 450 km/h. bout 2 h lter, they re 800 km prt. Determine the ngle between their pths, to the nerest degree. Lesson x 6.0 m 14.0 m 10. From window in n 9 prtment building, 22 the ngle of elevtion 8 m to the top of flgpole cross the street is 9. The ngle of depression is 22 to the bse of the flgpole. How tll is the flgpole, to the nerest tenth of metre? 15.0 m 11. bush pilot delivers supplies to remote cmp by flying 255 km in the direction N52 E. While t the cmp, the pilot receives rdio messge to pick up pssenger t villge. The villge is 85 km S21 E from the cmp. Wht is the totl distnce, to the nerest kilometre, tht the pilot will hve flown by the time he returns to his strting point? 12. cnoeist strts from dock nd pddles 2.8 km N34 E. Then she pddles 5.2 km N65 W. Wht distnce, nd in which direction, should second cnoeist pddle to rech the sme loction directly, strting from the sme dock? Round ll nswers to the nerest tenth of unit. 154 hpter 3 cute Tringle Trigonometry
44 3 hpter Tsk cute Tringles in First Ntions nd Métis ultures Inside Dkot tipi lckfoot tipis Structures mde from cute tringles re used in most cultures. The First Ntions nd Métis peoples who lived on the Priries used cutetringle structures to cook, dry met, nd trnsport goods nd people. cute tringles re lso the support for tipis. Tipis suit life bsed on the migrtion of the bufflo, becuse they re esily ssembled nd tken down. Tipis re still used for shelter on cmping trips nd for ceremonil purposes. The frme of tipi consists of wooden poles supported by n inner tripod. dditionl poles re lid in nd tied with rope. When the frme is complete, the covering is drwn over the frme using two dditionl poles. The interior of the tipi cretes roughly conicl shpe. The number of poles used is indictive of geogrphic re nd people, rnging from 13 to 21 poles.? Wht re the mesures of ll the sides nd ngles in ech tringle in the frme of tipi?. Design tipi. hoose the number of poles, the interior height, nd the dimeter of the bse.. Wht regulr polygon forms the bse of your tipi? Wht type of tringle is formed by the ground nd supporting poles? Explin how you know.. Determine the side lengths of the frme tringles. D. Determine the interior ngles of the frme tringles. E. Explin why your design is functionl. frme tringle Tsk hecklist Did you drw lbelled digrms for the problem? Did you show your work? Did you provide pproprite resoning? Did you explin your thinking clerly? hpter Tsk 155
45 3 Project onnection reting Your Reserch Question or Sttement wellwritten reserch question or sttement clrifies exctly wht your project is designed to do. It should hve the following chrcteristics: The reserch topic is esily identifible. The purpose of the reserch is cler. The question/sttement is focused. The people who re listening to or reding the question/sttement will know wht you re going to be reserching. good question requires thought nd plnning. elow re three exmples of initil questions or sttements nd how they were improved. Uncceptble Question/Sttement Is mthemtics used in computer technology? Wter is shred resource. Do driver s eduction progrms help teengers prllel prk? Why? too generl too generl too specific, unless you re going to generte your own dt cceptble Question/Sttement Wht role hs mthemtics plyed in the development of computer nimtion? Homes, frms, rnches, nd businesses est of the Rockies ll use runoff wter. When there is shortge, tht wter must be shred. Do driver s eduction progrms reduce the incidence of prking ccidents? Evluting Your Reserch Question or Sttement You cn use the following checklist to determine if your reserch question/sttement is effective. 1. Does the question/sttement clerly identify the min objective of the reserch? fter you red the question/sttement to few clssmtes, cn they tell you wht you will be reserching? 2. re you confident tht the question/sttement will led you to sufficient dt to rech conclusion? 3. Is the question/sttement interesting? Does it mke you wnt to lern more? 4. Is the topic you chose purely fctul, or re you likely to encounter n issue, with different points of view? 156 hpter 3 cute Tringle Trigonometry
46 Project exmple Writing reserch question Srh chose the chnges in popultion of the Western provinces nd the territories over the lst century s her topic. elow, she describes how she determined tht her reserch question for this topic is effective. Srh s Question My question is, Which Western province or territory grew the fstest over the lst century nd why? I will use 1900 to 2000 s the time period. I evluted my question using the reserch question checklist, nd I feel tht it is good one. Here is why: 1. My question tells wht I pln to do: I red my question to three friends, nd they ll described wht I hd in mind. 2. I m confident tht there is lot of dt vilble on popultions, nd tht there is lot of historicl informtion vilble on why the popultions chnged. 3. I m relly interested in history, but I don t know enough bout how the West nd North grew nd why. I ll find out lots of new things. Whtever I find out should lso interest some of my clssmtes, s it s bout where we live. 4. I expect tht I will find severl different points of view on why popultions grew, nd I hope tht I will be ble to conclude which fctors were most importnt. Your Turn. Write reserch question for your topic.. Use the checklist to evlute your question. djust your question s needed.. Mke n ppointment to discuss the pros nd cons of your reserch question with your techer. e prepred to discuss your pln for collecting the dt you will need to come to conclusion. djust your question s needed. hinese immigrnts nd workers rrive t Willim Hed Qurntine Sttion, ritish olumbi, in Wht fctors ffected where they chose to settle? This photogrph shows potsh mine in Ssktchewn. How do resources ffect popultion growth? Project onnection 157
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