MBF 3C Unit 2 Trigonometry Outline

Transcription

1 Dy MF 3 Unit 2 Trigonometry Outline Lesson Title Speifi Expettions 1 Review Trigonometry Solving for Sides Review Gr Review Trigonometry Solving for ngles Review Gr Trigonometry in the Rel World Sine Lw osine Lw hoosing between Sine nd osine Lw Rel World Problems More Rel World Problems Review Dy 10 Test Dy TOTL DYS: solve problems, inluding those tht rise from rel-world pplitions (e.g., surveying, nvigtion), by determining the mesures of the sides nd ngles of right tringles using the primry trigonometri rtios; 2.2 verify, through investigtion using tehnology (e.g., dynmi geometry softwre, spredsheet), the sine lw nd the osine lw (e.g., ompre, using dynmi geometry softwre, the rtios /sin, b/sin, nd in tringle while drgging one /sin of the verties); 2.3 desribe onditions tht guide when it is pproprite to use the sine lw or the osine lw, nd use these lws to lulte sides nd ngles in ute tringles; 2.4 solve problems tht rise from rel-world pplitions involving metri nd imperil mesurements nd tht require the use of the sine lw or the osine lw in ute tringles.

2 Unit 2 Dy 1: Trigonometry Finding side length MF 3 Minds On Desription This lesson reviews Trigonometry Mteril from the Grde 10 ourse speifilly solving sides of tringles using the three trigonometri rtios. Whole lss Disussion Write the mnemoni SOHHTO on the bord nd see wht the students n rell from lst yer s mteril. Use this to re-introdue the three primry trigonometri rtios; Sine, osine nd Tngent Sine Opposite over Hypotenuse SOH osine djent over Hypotenuse H Tngent Opposite over djent TO Mterils LM2.1.1 Sientifi lultor ssessment Opportunities tion! Use the following digrm to id in identifying right tringle. b opposite side to (n be djent side to ) hypotenuse (lwys opposite the right ngle) opposite side to (n be djent side to ) Note the differenes in the results if we onsider we re looking from insted: sin b os tn b These re the primry trigonometri rtios when we look t : sin os tn opposite side to length of side hypotenuse length of side djent side to length of side b hypotenuse length of side opposite side to length of side djent side to length of side b Word Wll: Sine osine Tngent Perform Think loud on the first exmple to find the missing side.

3 . Exmple 1: Find the length of side. 12 m 25 Sript for Think loud: We wnt to find side. First, I wnt to exmine the tringle to determine wht informtion is given to us. We hve nd the hypotenuse. Sine, is the side opposite to, we will need to use the sine trigonometri rtio whih is: opposite side to sin hypotenuse length of side length of side Next, I wnt to put in the informtion tht we know into our eqution. We ll reple with 25 nd with 12. So we get, sin I m going to use the lultor to find the deiml vlue of the rtio for sin 25. I wnt to mke sure the lultor is in the proper mode. I wnt it to be in deiml mode. Oky now the lultor shows tht the rtio is worth nd this we n reple sin 25 in the eqution, giving us To solve for, I wnt to multiplying both sides by x x 12 nd the result beomes, the length of side is pproximtely 5.1 m long. sk students to do exmple 2 with prtner. Emphsize with the lss to mke sure they selet the pproprite trigonometri rtio.

4 Exmple 2: Find the length of side m Tke up exmple 2 with the lss. Give exmple 3 nd disuss with the lss possible strtegies to solve for. One possible strtegy is to use (50 0 sum of the ngles in tringle) nother possible strtegy is s shown below. sk student to individully try exmple 3 on their own. Hve students shre their solutions. Exmple 3: Find the length of side. 40 os dj side to hyp os os the length of side is pproximtely 12.3 m long. 10 m tn tn opposite side to djent side to length of side b length of side 10 lternte solution: tn opp side to dj side to tn b tn x x To solve for : divide both sides by the length of side is pproximtely 11.9 m long the length of side is pproximtely 11.9 m long.

5 onept Prtie Skill Drill Home tivity or Further lssroom onsolidtion Students omplete LM2.1.1.

6 MF3 LM2.1.1 Nme: Dte: Digrms re not drwn to sle. 1. sed on the following digrm use the vlues given to find the missing/indited side: () 55, 25 m find (b) 65, 32 m find b b () 15, 42 m find b (d) 35, 55 m find 2. sed on the following digrm use the vlues given to find the missing/indited side: () 75, b 52 m find (b) () 64, 23 m find b 18, 24 m find b b (d) 31, b 58 m find 3. Given the following digrm solve for the lengths of the missing sides. 250 m Given the following digrm solve for the lengths of the missing sides. 125 m 65

7 MF3 LM2.1.1 Solutions: (Note: nswers should be within deiml ple depending on ury of numbers used.) 1. () 20.5m (b) b 13.5m () b 10.9m (d) 45.1m 2. () 194.1m (b) b 11.2m () b 7.8m (d) 96.5m 3. b 116.6m 275.8m m 137.9m

8 Unit 2 Dy 2: Trigonometry Finding ngle Mesure Desription This lesson reviews Trigonometry Mteril from the Grde 10 ourse speifilly solving ngles in tringles using the three trigonometri rtios. MF 3 Mterils Sientifi lultor LM2.2.1 Minds On Whole lss Disussion Drw right ngled tringle on the bord (s shown). Pose the question to the students: Given the sides how would you find the mesure of the missing ngles? ssessment Opportunities Students should be ble to relte the trigonometri rtios lerned yesterdy to begin to mke the onnetion to finding ngles. Hve disussion tht then leds into the lesson shown below tion! Whole lss Guided Instrution sk students to reflet on the primry trigonometri rtios for nd to onnet it to the bove tringle : 5 sin os tn opposite side to length of side hypotenuse length of side djent side to length of side b hypotenuse length of side opposite side to length of side djent side to length of side b Speifilly for the bove tringle: 5 12 sin os sin os tn tn 12 5 Show students how to use the lultor to solve for the ngle in ny of the bove ses by essing the inverse of eh of the trigonometri funtions It relly didn t mtter whih trigonometri rtio we hose to use in order to find the orret ngle. Usully ngles re rounded to the nerest degree. Therefore is pproximtely 23. NOTE to students:: Every lultor is different. Some require you to enter the vlue first nd then do the 2 nd button nd then the Sin button, others require you to hit the 2 nd button, then sin before entering the vlue. Test to see whih order your lultor uses. Do exmple 1 with the students.

9 Exmple 1: Find the mesure of. 12 m 5 m First we need to exmine the tringle to determine wht we should do: We hve the side opposite nd the hypotenuse nd we need to find the mesure of. This would be esiest using the sine trigonometri rtio. opposite side to sin hypotenuse length of side sin length of side 5 sin 12 sin Use your lultor to find (sin 1 ) is pproximtely 25. sk students to work with prtner to do exmple 2 nd 3. Exmple 2: Find the mesure of. 12 m 15 Exmple 3: Find the mesure of. os os os dj side to hyp os is pproximtely tn tn tn opp side to dj side to b tn is pproximtely 40.

10 Some students might sy using different trig rtio nd others might omment on the sum of the ngles in tringle. onept Prtie Skill Drill Home tivity or Further lssroom onsolidtion Students omplete LM2.2.1

11 MF3 LM2.2.1 Nme: Dte: Digrms re not drwn to sle. Round ngle mesures to the nerest degree. The side length nswers should be rounded to one deiml ple. 1. sed on the following digrm, use the vlues given to find the missing/indited side: () 58 m, 124 m find (b) b 75 m, 215 m find () b 64 m, 225 m find b (d) 45 m, 238 m find 2. sed on the following digrm, use the vlues given to find the missing/indited side: () 55 m, b 137 m find (b) () 235 m, b 68 m find 212 m, b 100 m find b (d) 30 m, b 285 m find 3. Using the digrm below on the left solve for the mesure of the missing ngles. 235 m 123 m 275 m 65 m 4. Using the digrm bove on the right solve for both the mesure of the missing ngles nd the length of the missing side. Solutions: 1. () 28 (b) 70 () 17 (d) () 22 (b) 74 () 25 (d) , , 31, b 142.8m

12 Unit 2 Dy 3: Trigonometry pplitions Desription This lesson ontinues the use of Trigonometri Rtios from the lst two dys but pplies them to rel world problems. MF 3 Mterils hrt pper mrkers Minds On Pirs Prtie ssessment Opportunities Write the following problem on the bord: You re out in field flying your kite. You hve just let out ll 150 m of your kite string. You estimte tht the kite is t n ngle of elevtion from you of bout 20. n you lulte the height of your kite bove the ground? (Hint: try drwing digrm.) Students refer to word wll of the trig rtios s needed. Let the students work on this problem for while nd then help them with the solution: 150 m 20 Drw the sketh on the bord. Exmining the tringle, we see tht we hve n ngle nd the hypotenuse, so we need to find the side opposite the given ngle. This sounds like the sin trigonometri rtio. opposite side to sin hypotenuse length of side sin length of side k sin k is the height of the kite we wnt to find. k Solving for k by multiplying both sides by 150 gives. k the kite is pproximtely 51 m bove the ground. Tody s lesson will inlude more rel world problems using trigonometri rtios.

13 tion! Smll Groups Plemt Provide eh group with one of the following problems (LM2.3.1) to omplete on plemt or hrt pper. Problem 1: While wlking to shool you pss brn with silo. Looking up to the top of the silo you estimte the ngle of elevtion to the top of the silo to be bout 14. You ontinue wlking nd find tht you were round 40 m from the silo. Using this informtion nd your knowledge of trigonometri rtios lulte the height of the silo m Possible Solution: tn opposite side to djent side to length of side b tn length of side s tn s s represents the height of the silo s the silo is pproximtely 10 m high. Problem 2: silbot is pprohing liff. The ngle of elevtion from the silbot to the top of the liff is 35. The height of the liff is known to be bout 2000 m. How fr is the silbot wy from the bse of the liff? 2000 m 35

14 Possible solution tn tn tn35 opposite side to djent side to length of side b length of side 2000 s represents the distne between the silbot nd the liff. s 2000 s s 2000 s the silbot is pproximtely 2856 m wy from the liff. (or lmost 3 km) Problem 3: silbot tht is 2 km due west of lighthouse sends signl to the lighthouse tht it is in distress. The lighthouse quikly signls resue plne tht is 7 km due south of the lighthouse. Wht heding from due north should the plne tke in order to interept the troubled silbot? 2 km 7 km Possible Solution: opposite side to tn djent side to length of side tn length of side b 2 tn 7 tn the plne should tke heding of bout 16 west of north to interept nd resue the silbot.

15 onsolidte Debrief Smll Groups Presenttion Ple the smll groups together tht hve the sme problem to hek eh others work. Eh of the groups prepres for the presenttion. Rndomly selet one of the groups for eh problem to present to the lss. pplition Home tivity or Further lssroom onsolidtion Students omplete LM2.3.2

16 MF3 LM2.3.1 Trigonometry pplition Problems Problem 1: While wlking to shool you pss brn with silo. Looking up to the top of the silo you estimte the ngle of elevtion to the top of the silo to be bout 14. You ontinue wlking nd find tht you were round 40 m from the silo. Using this informtion nd your knowledge of trigonometri rtios lulte the height of the silo m Problem 2: silbot is pprohing liff. The ngle of elevtion from the silbot to the top of the liff is The height of the liff is known to be bout 2000 m. How fr is the silbot wy from the bse of the liff? 2000 m 35

17 MF3 LM2.3.1 Trigonometry pplition Problems (ontinued) Problem 3: silbot tht is 2 km due west of lighthouse sends signl to the lighthouse tht it is in distress. The lighthouse quikly signls resue plne tht is 7 km due south of the lighthouse. Wht heding from due north should the plne tke in order to interept the troubled silbot? 2 km 7 km

18 MF3 LM2.3.2 Nme: Dte: Round s to whole degrees. Length nswers should be rounded to 1 deiml ple nd inlude units. 1. The top of lighthouse is 100 m bove se level. The ngle of elevtion from the dek of the silbot to the top of the lighthouse is 28. lulte the distne between the silbot nd the lighthouse. 100 m n rher shoots nd gets bulls-eye on the trget. From the rher s eye level the ngle of depression to the bulls-eye is 5. The rrow is in the trget 50 m below the rher s eye level. lulte the distne the rrow flew to hit the trget (the dotted line) m For the following questions you will need to rete your own digrms. Drw them refully nd refer to the written desription to ensure you find the orret solution. 3. Two islnds nd re 3 km prt. third islnd is loted due south of nd due west of. From islnd the ngle between islnds nd is 33. lulte how fr islnd is from islnd nd from islnd. 4. The foot (bottom) of ldder is pled 1.5 m from wll. The ldder mkes 70 ngle with the level ground. Find the length of the ldder. (Round to one deiml ple.) 5. tow truk rises the front end of r 0.75 m bove the ground. If the r is 2.8 m long wht ngle does the r mke with the ground? 6. onstrution engineer determines tht stright rod must rise vertilly 45 m over 250 m distne mesured long the surfe of the rod (this represents the hypotenuse of the right tringle). lulte the ngle of elevtion of the rod. Solutions: m m 3. Distne to : 1.6 km Distne to : 2.5 km m

19 Unit 2 Dy 4: Trigonometry Sine Lw MF 3 Minds On Desription This lesson introdues the Sine Lw to the students. Whole lss Disussion Drw the following digrm on the bord: 10 x Mterils omputer Lb with Geometer s SkethPd. LM2.4.1, ssessment Opportunities tion! Pose the following question to the students: With wht we know from the digrm, n we find the vlue of x? sk the students, if more informtion ws provided like the following digrm if tht would help them solve for x? Led students through the following solution: h 0 sin x 10 h h 8. 66m h 0 Now, sin 50 x 8.66 x 0 sin 50 x 11.3 m Observtion: h 10sin60 0 nd h 11.3sin50 0 in this se sin sin Pose the question to them: Is this true ll the time? Pirs Investigtion Students use Geometer s Skethpd to explore the sine lw following LM Intended Response: No, too hrd for us! The students re disovering the sine lw using numeril vlues. If the shool uses LnShool or some brodst pbility for the teher then the teher ould brodst the steps first nd let the students try it fterwrd.

20 Whole lss Guided Instrution Demonstrte to students how to use the Sine Lw to solve tringles. Exmple 1: In Δ, given tht 48, 25, nd side (nmed s ) 36 m. Find the length of nd orret to 1 deiml ple. (Help by drwing the digrm inluded below.) m 25 ording to the Sine Lw we need rtio of the sine of n ngle nd its orresponding side, urrently we don t hve this, however, we do hve nd so we re ble to solve for ( ) 43 We hve 36 m nd we need to find nd. So using the Sine Lw sin b sin sin 36 sin 43 sin 48 sin 25 we hve: Solving seprtely: 36 sin 43 sin 48 nd 36 sin 43 sin 25 Here, multiply both sides by sin 48 nd here, multiply both sides by sin 25 Gives: sin 48 nd sin 25 sin 43 sin 43 Finishing off using lultor: 39.2 m nd 22.3 m Hve students working with prtner solve exmple 2. Exmple 2: Solve for the vlue of h in the following digrm: h m

21 Possible Solution: Let s lbel the digrm t the pek nd on the bse. Thus, in sin b sin sin We get: 68 b sin 81 sin 43 sin 56 We n solve for either of b or nd then use the primry trigonometri rtios to omplete the solution for h. 68 b sin 81 sin 43 or 68 sin 81 sin 56 Solving for the hosen side following the sme steps s bove we get: b 47.0 m or 57.1 m Finlly to solve for h use the sine trigonometri rtio. Sin 56 h 47 or sin 43 h h 38.9 m or h 38.9 m 57 onsolidte Debrief Whole lss Disussion Summrize when to use the Sine Lw. The Sine Lw: Using the sme tringle bove you ould onstrut nother perpendiulr line from or to the opposite side nd rete similr expression for sin nd its orresponding side. In generl the Sine Lw tkes the form: sin sin sin b or sin b sin sin Emphsize tht the two equl signs onstitutes three equtions. onept Prtie Home tivity or Further lssroom onsolidtion Students omplete LM2.4.2.

22 MF 3 LM Nme: Dte: Investigtion: Sine Lw Geometer s Skethpd 1. Lod Geometer s Skethpd. 2. Strt with new doument (defult). 3. Selet the Strightedge Tool (4 th button down the toolbr) 4. Drw three lines mking sure tht eh new line strts from previous line nd tht the lst point returns to the first ompleting the tringle. (shown right) 5. Swith to the seletion tool (1 st button on the toolbr) 6. Selet nd right-lik on eh vertex nd from the short-ut menu selet Show Lbel (lso shown right) 7. Next selet ny line nd from the Mesure menu (or from the right-lik short-ut) selet Length. This should disply m (shown) 8. Repet Step 6 for the other lines, mking sure to unselet before seleting new line. (If nything else is seleted length my not pper on the menu.) 9. Next selet in the following order the verties:, then then lik the Mesure menu nd hoose ngle. This should disply m nd the mesure of tht ngle. 10. Now repet Step 8 but for ngles nd. (shown) 11. If you selet ny point you n drg the point to new lotion nd ll of the mesurements updte utomtilly. (You n lso selet nd move n entire line.) 12. Try this nd djust the position of the tringle to leve more room below our mesurements. 13. We will now dd some lultions nmely the vlues for the Sine Lw: b sin sin sin 14. To do this selet the Mesure menu nd selet lulte. new dilogue box ppers (shown right) where we will enter our lultion. 15. First lik on the mesurement for side (in this se it is m ), then lik on the division sign

23 nd type si for the sine funtion, next lik on the mesurement for (in this se it is m (depending on the size of your tringle you will see different results.) lik OK 16. This will dd new mesurement to your doument, repet step 15 for side b nd side. For side b use m nd sin(m ) for side use m nd sin(m ). lultions re shown in the bottom digrms. 17. Now hnge the position of your verties; this will hnge the lengths nd ngles in your tringle mke note of wht hppens to ll three of the lultion boxes b sin sin sin for the Sine Lw:. (two vritions shown below) 18. Next rete three more lultions for the other version of the Sine Lw: sin sin sin b (shown right) 19. Experiment with more positions of the tringle verties. 20. Notie tht the set of three vlues in either version of the Sine Lw remin the sme. This shows tht the rtio of ny side to the sine of the orresponding ngle in tringle remins equl to the rtio of ny other side to the sine of the orresponding ngle. Either sin sin sin b or b sin sin sin

24 MF3 LM2.4.2 Nme: Dte: 1. Solve for the given vrible (orret to 1 deiml ple) in eh of the following: () b 75 (b) () sin 35 sin 40 sin 75 sin 48 sin 55 sin For eh of the following digrms write the eqution you would use to solve for the indited vrible: () (b) () b m m m 3. Solve for eh of the required vribles from Question #2. 4. For eh of the following tringle desriptions you should mke sketh nd then find the indited side rounded orretly to one deiml ple. () (b) () (d) (e) (f) In Δ, given tht 57, 73, nd 24 m. Find the length of In Δ, given tht 38, 56, nd 63 m. Find the length of In Δ, given tht 50, 50, nd 27 m. Find the length of In Δ, given tht 23, 78, nd 15 m. Find the length of In Δ, given tht 55, 32, nd 77 m. Find the length of In Δ, given tht 14, 78, nd 36 m. Find the length of Solutions: 1. () 8.9 units (b) 50.0 units () 90.2 units () (b) sin 53 sin b sin 35 sin 70 () 14.2 sin 15 sin () 29.1 m (b) 38.7 m () 52.5 m 4. () 30.0 m (b) 52.4 m () 34.7 m (d) 6.0 m (e) 49.8 m (f) m

25 Unit 2 Dy 5 - Trigonometry - osine Lw MF 3 Minds On Desription This lesson introdues the osine Lw to the students. Whole lss Disussion Mterils LM2.5.1 Sientifi lultor ssessment Opportunities Drw the following digrm on the bord: 12 m 26 m sk students to solve for in the tringle bove using the Sine Lw: 60 Results: sin sin sin 60 tion! The solution is stlled t this point sine eh prt of the rtio hs some missing informtion. We nnot solve the tringle. We need to develop new formul this formul is lled the osine Lw Whole lss Guided Instrution Pose to the students: Hven t you lwys dremed bout using the Pythgoren Theorem for ll tringles? Rell the most fmous Pythgoren tringle For Δ, 4 6 h 4 2 x D 5 x h + x 4 nd h 2 + ( 5 x ) h h ( 5 x) x 1 2 From nd x 6 (5 x x 6 (5 10x + x x x 10x os 6 ) 2 ( ) b b os Is this true for ll tringles? ) os 4 x 4 os x

26 Guide students through exmple 1 nd 2 to solve for missing side nd missing ngle using the osine Lw. Exmple 1: Now we n solve for from our first problem (redrwn below) b 2 2b os 2 (26) 2 + (12) 2 2 (26) (12) os (0.5) 12 m m m Exmple 2: In Δ, given 7 m, 8 m nd 10 m. Find the mesure of to the nerest degree. (Help by drwing the digrm inluded below.) 10 m 8 m 7 m We re sked to find the mesure of to the nerest degree so we should use the osine Lw formul tht is most pproprite: 2 b b os Filling in wht we know gives: (7) 2 (8) 2 + (10) 2 2(8)(10) os os Rerrnging: 160 os os os 115 Finlly: 115 os 160 os Use your lultor to solve for onsolidte Debrief to the nerest degree is 44 Whole lss Disussion sk students to set up the equtions using the osine Lw for questions #1, nd #1b on LM Verify tht they hve the orret set up. onept Prtie Home tivity or Further lssroom onsolidtion Students omplete LM2.5.1.

27 MF3 LM2.5.1 Nme: Dte: 1. For eh of the following digrms write the eqution you would use to solve for the indited vrible: () (b) () 28.4 m 26 m 23.6 m m 14.2 m 22.4 m m 2. Solve for eh of the required vribles from Question #1. 3. For eh of the following tringle desriptions you should mke sketh nd then find the indited vlue. () (b) () (d) In Δ, given tht 24 m, 34 m, nd 67. Find the length of In Δ, given tht 15 m, 8 m, nd 24. Find the length of In Δ, given tht 10 m, 9 m, nd 48. Find the length of In Δ, given tht 24, 18.6 m, nd 13.2 m. Find the length of (e) In Δ, given tht 9 m, 12 m, nd 15 m. Find the mesure of. (f) In Δ, given tht 18.4 m, 9.6 m, nd 10.8 m. Find the mesure of. Solutions: 1. () 2 (36) 2 + (26) 2 2(36)(26) os 53 (b) (28.4) 2 (23.6) 2 + (33.2) 2 2(23.6)(33.2) os () 2 (22.4) 2 + (14.2) 2 2(22.4)(14.2) os () 29.1 m (b) 57 () 23.2 m 3. () 33.1 m (b) 8.4 m () 7.8 m (d) 8.5 m (e) 53 (f) 24

28 Unit 2 Dy 6 Trigonometry pplying the Sine nd osine Lw Desription This lesson hs students solving tringles by hoosing between the Sine Lw nd the osine Lw. MF 3 Mterils LM2.6.1,2.6.2 Minds On tion! onsolidte Debrief Whole lss Four orners Post four signs, one in eh orner lbelled Sine Lw, osine Lw, Pythgoren Theorem, Trigonometri Rtio (SOHHTO). Provide eh of the students with one of the tringles on LM Instrut the students to mke deision s to whih method they would use to solve the missing ngle or side nd to stnd in the orner where it is lbelled. One students re ll pled hve the students disuss mongst themselves to onfer tht they hve seleted n pproprite method. llow students to move to different lotion fter disussion. sk one representtive from eh orner to explin why their tringle(s) would be best solved using tht prtiulr method. sk students to omplete Fryer model (LM2.6.2) for their method. dd to word wll or lss mth ditionry. Pirs Prtie sk students to individully solve the tringle they were given for the previous tivity. Then with n elbow prtner hek eh others work. Possible solutions for the tringles on tehers opy of the Four orners. Whole lss Disussion lrify ny problems from the pirs solving for their missing side or ngle. Highlight wht the phrse solving tringle mens (solve for ll sides nd ngles in tringle). Pirs Prtie sk pirs to trde questions nd solve for the remining prts of their tringle. ssessment Opportunities LM2.6.1 nswers: 1.Sine Lw 2.osine Lw 3.osine Lw 4.Sine Lw 5.Pythgoren Theorem 6.Trig Rtio Support students with the hrteristis. My wnt to inlude items like: 3 sides you need to use osine Lw to solve for ngles. 2 sides nd 1 ngle it depends on where the ngle is loted if the ngle is between the two sides (ontined) then you need to use the osine Lw usully to find lst side. ii) if the ngle is not ontined then the Sine Lw n be used usully to find nother ngle. 1 side nd 2 ngles then you need to use the Sine Lw, usully to find side. pplition Home tivity or Further lssroom onsolidtion Students omplete LM2.6.3

29 MF3 LM2.6.1 Four orners Tringles 1) 2) m 14 m m 3) 4) 28 m 23 m 33 m 26 m m 5) 6) 123 m 125 m 65 m 45 m

30 MF3 LM2.6.1 Four orners Tringles (Teher) 1) 2) m 43 First we need to find we n do this using the sum of the ngles in tringle Therefore, 81 Now we n solve for using the Sine Lw b sin sin 38 sin 56 sin sin sin m Solving for the other sides: Sine Lw vs. osine Lw b sin sin b 2 2b os 38 sin 81 sin 43 2 (31.9) 2 + (38) 2 2(31.9)(38) os m m 2 b b os 2 (14) 2 + (22) 2 2(14)(22) os m Solving for the other ngles. sin sin b sin 80 sin sin sin sin sin this gives us 65 (i.e ) 38 sin sin ( ) m m

31 MF3 LM2.6.1 Four orners Tringles (Teher) 3) 4) 23 m 33 m 28 m 26 m 37 m 2 b b os (28) 2 (23) 2 + (33) 2 2(23)(33) os os Rerrnging: 1518 os os os 1518 os Solving for the other ngles: sin sin b sin 57 sin sin sin sin sin this gives us 79 (i.e ) 53 sin sin sin sin sin 53 sin sin sin Solving for the other sides b sin Sine Lw vs. osine Lw sin b os b 37 sin 93 sin 53 b 2 (26) 2 + (37) 2 2(26)(37) os sin sin b b ( ) 37 b b b 46.3 m b 46.3 m

32 MF3 LM2.6.1 Four orners Tringles (Teher) 5) 6) 123 m 125 m 65 m Solving for ngles: 123 tn 65 tn m 125 tn 45 tn Solving for the other ngle nd side m

33 MF3 Nme: LM2.6.2 Fryer Model Dte: Definition hrteristis Exmples Non-exmples

34 MF3 LM2.6.3 Nme: Dte: Round s to whole degrees; length nswers should be rounded to 1 deiml ple nd inlude units. 1. For eh of the following digrms write the eqution you would use to solve for the indited vrible: () (b) () b m 14.2 m m 40 m (d) (e) (f) 10.7 m 21.3 m 9.5 m 14.2 m 12.4 m 9 m m Solve for eh of the required vribles from Question #1. 3. For eh of the following tringle desriptions you should mke sketh nd then ompletely solve eh tringle. () In Δ, given tht 38, 85, nd 32 m. (b) In Δ, given tht 24, b 12.5 m, nd 13.2 m. () In Δ, given tht 17 m, b 18 m, nd 26 m. (d) In Δ, given tht 52, 47, nd 25 m. (e) In Δ, given tht 43, 73, nd b 19 m. (f) In Δ, given tht 32 m, b 30 m, nd 28 m. Solutions: 1. () 2 (40) 2 + (25) 2 2(40)(25) os 20 (b) () 46 sin 20 sin b sin 38 sin 67 (d) (10.7) 2 (9.5) 2 + (12.4) 2 2(9.5)(12.4) os (e) 2 (10) 2 + (9) 2 2(10)(9) os 66 (f) sin sin () 18.6 m (b) 21.2 m () 20.5 m (d) 57 (e) 10.4 m (f) 40 (Note: the following nswers hve been listed in the optiml order for solving the tringle. If you did not solve your tringle in this order your ngle mesurements should be within 1 due to rounding differenes; side length vlues should be urte.) 3. () 57, 19.8 m, b 26.9 m (b) 5.4 m, 70, 86 () 41, 43, 96 (d) 81, b 23.2 m, 31.3 m (e) 64, 25.0 m, 26.6 m (f) 67, 60, 53

35 Unit 2 Dy 7: Trigonometry Solving Rel-World Problems Desription This lesson hs students solving rel-world problems using the Sine Lw nd the osine Lw. MF 3 Mterils LM2.7.1 Minds On tion! Pirs Interpreting Problem E Write the following exmple on the bord: sk students to develop digrm to illustrte 2000 km Problem 1:. Dvid wnts to go to Toronto from Edmonton, but he took the wrong rod nd ended up in higo insted. Upon relizing his diretionl mistke, Dvid drove from higo to Toronto. If the ngle t Toronto is 45, the ngle t higo 95 0 is 95, nd the distne from Edmonton to Toronto is 2000 km, how muh further did Dvid drive thn neessry? Tke up the digrm with the lss. Pirs Prtie sk students to finish solving the bove problem nd Problem 2. Solution: The E n be found using the sum of the ngles in tringle: E 40 ssessment Opportunities 45 0 T Now use Sine Lw to find the remining two sides: e t sin E sin T sin e t 2000 sin 40 sin 45 sin 95 rek this into two seprte equtions nd solve for e nd t. e 2000 t 2000 sin 40 sin 95 sin 45 sin e sin 40 t sin 45 sin 95 sin e t e km t km So Dvid drove totl of km. he drove km more thn ws neessry.

36 Problem 2: Jill nd her friends built n outdoor hokey rink. Their hokey gol line is 5 feet wide. Jill shoots puk from point where the puk is 5 yrds from one gol post nd 6 yrds from the other gol post. Within wht ngle must Jill mke her shot to hit the net? Solution: 5 feet 5 yrds 6 yrds If we mke the position where Jill is stnding nd the golposts nd then in this se we n use the osine Lw to solve for the ngle. 2 b b os (5) 2 (15) 2 + (18) 2 2(15)(18) os os os os os os Jill must shoot within n ngle of bout 14 to hit the net. onsolidte Debrief Pir/Group Shre sk pir of students to shre their solution with nother pir. s whole lss disuss ny problems nd shre model solutions.. pplition Home tivity or Further lssroom onsolidtion Students omplete LM2.7.1

37 MF3 LM2.7.1 Nme: Dte: If digrms re not inluded in ny of the following questions it is dvisble to sketh digrm to id in your solution to the problem. Round s to whole degrees; length nswers should be rounded to 1 deiml ple nd inlude units. 1. sqush plyer hits the bll 2.3 m to the side wll. The bll rebounds t n ngle of 100 o nd trvels 3.1 m to the front wll. How fr is the bll from the plyer when it hits the front wll? (ssume the plyer does not move fter the shot.) 2. smokestk,, is 205m high. From two points nd D on the sme side of the smokestk s bse, the ngles of elevtion to the top of the smokestk re 40 o nd 36 o respetively. Find the distne between nd D. (Digrm inluded.) 205 m D 3. Trin nd Mzheer re stnding on the sme side of Red Mple tree. The ngle of elevtion from Mzheer to the tree top is 67 nd the ngle of elevtion from Trin to the tree top is 53. If Mzheer nd Trin re 9.3 feet prt nd Mzheer is loser to the tree thn Trin, how tll is the tree? 4. Two rods seprte from villge t n ngle of 37. Two ylists leve the villge t the sme time. One trvels 7.5 km/h on one rod nd the other trvels 10.0 km/h on the other rod. How fr prt re the ylists fter 2 hours? 5. pilot is flying from Thunder y, Ontrio to Dryden, Ontrio, distne of pproximtely 320 km. s the plne leves Thunder y, it flies 20 off-ourse for extly 80 km. () fter flying off-ourse, how fr is the plne from Dryden? (b) y wht ngle must the pilot hnge her ourse to orret the error? Solutions: m m feet km 5. () km (b) pproximtely 26 turn towrds Dryden.

38 Unit 2 Dy 8 - Trigonometry Solving Rel World Problems ontinued Desription This lesson hs students solving more rel-world problems using the Sine Lw nd the osine Lw. MF 3 Mterils Minds On Individul reting Digrm Write the following exmple on the bord. sk students to rete piture for the following problem. lrify the term ngle of elevtion. ssessment Opportunities Word wll: ngle of elevtion Jillin stood t distne dmiring mgnifient Dougls Fir. Jillin mesured the ngle of elevtion to the top of the tree nd found it to be 15. Jillin then wlked 31.4 feet loser to the tree. This time the ngle of elevtion to the top of the tree ws 17. lulte the height of the tree to the nerest tenth of metre. tion! Whole lss Shring sk for students to shre their pitures to the lss. One digrm is estblished, disuss strtegies to solve the problem. Individul Prtie sk students to individully solve the problem feet D This problem will require few steps to omplete. Exmining the problem we note tht there re two min tringles: Δ nd ΔD. We need to find the height of the tree ( in Δ) but to find the height of the tree we need more informtion bout Δ. So we need to work in ΔD first. We n use the informtion we hve to find D sine D mkes stright line (180 ) nd we hve 17 we n find tht D 163 ( ) nd finlly D 2. So in order to find some informtion bout Δ in this se the hypotenuse we will use the Sine Lw to solve for. In ΔD we hve: sin d sin D 31.4 d sin 2 sin 15

39 d 31.4 sin 15 sin 2 d d feet This vlue is the length of the hypotenuse of Δ. So now we n use the primry trigonometri rtio for Sine to solve for the height of the tree. In Δ we hve: sin opposite side to hypotenuse sin sin 17 tree tree x sin 17 tree x tree onsolidte Debrief The tree is bout 68 feet tll. Whole lss Review ny other exmples from the lst dy s homework ould be done to help students before giving out tody s homework. lso ny extr time vilble ould be used for Quiz or review of ny other homework from previous lessons in this unit. pplition onept Prtie Home tivity or Further lssroom onsolidtion Students omplete LM

40 MF3 LM2.8.1 Nme: Dte: If digrms re not inluded in ny of the following questions it is dvisble to sketh digrm to id in your solution to the problem. Round s to whole degrees; length nswers should be rounded to 1 deiml ple nd inlude units. 1. To lulte the height of tree, Mrie mesures the ngle of elevtion from point to be 34. She then wlks 10 feet diretly towrd the tree, nd finds the ngle of elevtion from the new point to be 41. Wht is the height of the tree? 2. To mesure the distne from point to n inessible point, surveyor piks out point nd mesures to be 71. He moves to point, distne of 56 m from point, nd mesures to be 94 How fr is it from to? (Digrm below.) 3. rdr trking sttion lotes n oil tnker t distne of 7.8 km, nd silbot t distne of 5.6 km. t the sttion, the ngle between the two ships is 95. How fr prt re the ships? 4. Two islnds nd re 5 km prt. person took vtion from islnd nd trvelled 7 km to third islnd. t islnd the ngle seprting islnd nd islnd ws 34. While on this vtion the person deided to visit islnd. lulte how fr the person will hve to trvel to get to islnd from islnd. 5. The light from rotting offshore beon n illuminte effetively up to distne of 250 m. From point on the shore tht is 500 m from the beon, the sight line to the beon mkes n ngle of 20 with the shoreline. Wht length of shoreline is effetively illuminted by the beon? (i.e. solve for the length of D in the digrm below.) beon 500 m m shoreline 250 m D Solutions: feet m km km 5. HINT: When you solved for the ngle 43.2 tully is the vlue for ngle(s) D nd D (ΔD is isoseles sine D D nd D) nd the result 43.2 is too smll for Δ s (whih is tully hek sin vs sin 43.2 ) so the length of shoreline tht is effetively illuminted by the beon m.