Geometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 3 / 240. Slide 4 / 240. Slide 6 / 240.
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- Felicity Lynch
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1 Slide 1 / 240 Slide 2 / 240 New Jerse enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freel ville t nd is intended for the non-ommeril use of students nd tehers. These mterils m not e used for n ommeril purpose without the written permission of the owners. NJTL mintins its wesite for the onveniene of tehers who wish to mke their work ville to other tehers, prtiipte in virtul professionl lerning ommunit, nd/or provide ess to ourse mterils to prents, students nd others. Geometr Trigonometr of Right Tringles lik to go to wesite: Slide 3 / 240 Slide 4 / 240 Tle of ontents Pthgoren Theorem Similrit in Right Tringles Speil Right Tringles lik on Topi to go to tht setion Pthgoren Theorem Trigonometri Rtios Solving Right Tringles Return to the Tle of ontents ngles of levtion nd epression Lw of Sines nd Lw of osines re of n Olique Tringle Slide / 240 efore lerning out similr right tringles nd trigonometr, we need to review the Pthgoren Theorem nd the Pthgoren Theorem onverse. Slide 6 / 240 Rell tht right tringle is tringle with right ngle. hpotenuse leg leg The sides form tht right ngle re the legs. The side opposite the right ngle is the hpotenuse. The hpotenuse is lso the longest side.
2 Slide 7 / 240 Slide / 240 Pthgoren Theorem mple: In right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hpotenuse. Find the length of the missing side of the right tringle. leg2 + leg2 = hpotenuse2 or = 2 Is the missing side leg or the hpotenuse of the right tringle? Slide () / 240 Slide / 240 Solve for : mple: hpotenuse 22 = 2 1 = [This ojet is of pullthe t] right tringle? Is the missing side leg or the hpotenuse -1 is etrneous solution, distne n not equl negtive numer. = 1 Slide / 240 Slide () / mple: Find the length of the missing side of the right tringle. 20 mple: Is the missing side leg or the hpotenuse of the right tringle? = 2 = 1 Find the length of the missing side of the right tringle = 2 leg Is the missing side leg or the hpotenuse of the right tringle? 2 Find the length of the missing side of the right tringle. 20
3 Slide 11 / 240 Slide 11 () / The missing side is the of the right tringle. 1 The missing side is the of the right tringle. 6 hpotenuse hpotenuse leg leg 6 Slide / 240 Slide () / Find the length of the missing side. 2 Find the length of the missing side. 6 6 Slide 13 / 240 Slide 13 () / The missing side is the of the right tringle. 36 leg 1 hpotenuse hpotenuse leg 3 The missing side is the of the right tringle. 36 1
4 Slide 14 / 240 Slide 14 () / Find the length of the missing side Find the length of the missing side. 36 Slide 1 / 240 Slide 16 / 240 Rel World pplition 2 2 Solve using + =? 2 feet 7 feet Thus, the ottom of 2-foot ldder should e 7 feet from the wll. How fr up the wll will ldder reh? 2 The sfe distne of the se of the ldder from wll it lens ginst should e one-fourth of the length of the ldder. 2 feet? 7 feet The ldder will reh Slide 16 () / 240 feet up the wll sfel. Slide 17 / 240 Rel World pplition 2 2? 0 7 feet The ldder will reh 4 2 feet feet up the wll sfel. The dimensions of high shool sketll ourt re 4' long nd 0' wide. Wht is the length from one orner of the ourt to the opposite orner? 2 Solve using + =
5 Slide 17 () / N ourt is 0 feet wide nd the length from one orner of the ourt to the opposite orner is 6. feet. How long is the ourt? (Round the nswer to the nerest whole numer) = 2 6 = = The ourt is 7.7 feet 4.03 feet Rel World pplition Slide 1 / feet 11 feet 4 feet The dimensions of high shool sketll ourt re 4' long nd 0' wide. Wht is the length from one orner of the ourt to the opposite orner? Slide 1 () / 240 Slide 1 / 240 Pthgoren Theorem pplitions N ourt is 0 feet wide nd the length from one orner of the ourt to the opposite orner is 6. feet. How long is the ourt? 4.03 feet feet 11 feet (Round the nswer to the nerest whole numer) The Pthgoren Theorem n lso e used in figures tht ontin right ngles. 4 feet Slide 20 / 240 Slide 21 / 240 mple Find the perimeter of the squre. Rememer, in squre ll sides re ongruent. Psq = 4s note: efore finding the perimeter of the squre, we need to first find the length of eh side. Strt here: = 1 1 m 1 m
6 Slide 21 () / 240 Slide 22 / 240 Rememer, in squre ll sides re ongruent. Strt here: = 1 1 m The se of the tringle is given, ut we need to find the height of the tringle. = 1 2 h 13 feet 13 feet feet Slide 23 / 240 Slide 23 () / feet h feet 13 feet definition, the ltitude (or height) of n isoseles tringle is the perpendiulr isetor of the se. feet 13 feet 13 feet h feet definition, the ltitude (or height) of n isoseles tringle is the perpendiulr isetor of the se. feet Slide 24 / 240 Slide 24 () / 240 Tr this... Find the perimeter of the retngle. in P ret = 2l + 2w P in in ret in Tr this... Find the perimeter of the retngle. mple Find the re of the tringle. = 2l + 2w = 2 =6 Pret = 2(6) + 2() P = 2 inhes
7 Slide 2 / 240 Slide 2 () / Find the re of the retngle. 6 Find the re of the retngle. 0 squre feet 46 squre inhes feet f 17 ee t 46 squre feet 4 squre feet 4 squre feet 0 squre feet 46 squre inhes feet 17 fe e t 46 squre feet Slide 26 / 240 Slide 26 () / Find the perimeter of the squre. (Round to the nerest tenth) 2. m 2. m 36 m 2.6 m 36 m m m 2.6 m. m. m 7 Find the perimeter of the squre. (Round to the nerest tenth) Slide 27 / 240 Slide 27 () / inhes inhes 7 inhes Find the re of the tringle. Find the re of the tringle. 7 inhes 7 inhes h=4.in =24. squre inhes inhes
8 Slide 2 / 240 Slide 2 () / 240 Find the re of the tringle. Find the re of the tringle. 7 inhes 7 inhes 4 inhes 7 inhes 7 inhes h=6.7in =13.4 squre inhes 4 inhes Slide 2 / 240 Slide 30 / 240 If the squre of the longest side of tringle is equl to the sum of the squres of the other two sides, then the tringle is right tringle. If = +, then is right tringle mple Tell whether the tringle is right tringle. Rememer is the longest side If then Slide 31 / 240 Theorem If the squre of the longest side of tringle is greter thn the sum of the squres of the other two sides, then the tringle is otuse. is right tringle. 7 Rememer is the therefore is right tringle. longest side 2 F 7 2 F mple Tell whether the tringle is right tringle. 2 Slide 30 () / 240 is right tringle. If = +, then 24 onverse of the Pthgoren Theorem If > +, then is otuse
9 Slide 32 / 240 Slide 33 / 240 mple lssif the tringle s ute, right, or otuse. Theorem 1 17 Slide 33 () / 240 Slide 34 / 240 lssif the tringle s ute, right, otuse, or not tringle. mple lssif the tringle s ute, right, or otuse. = ute right otuse Sine 17 the tringle is ute. 1 not tringle If 2 < 2 + 2, then is ute. 13 If the squre of the longest side of tringle is less thn the sum of the squres of the other two sides, then the tringle is ute. 11 Slide 34 () / 240 Slide 3 / 240 right otuse not tringle ute ute 11 lssif the tringle s ute, right, otuse, or not tringle. otuse 1 11 right not tringle 3 6 lssif the tringle s ute, right, otuse, or not tringle.
10 Slide 3 () / 240 Slide 36 / 240 ute right otuse 3 lssif the tringle s ute, right, otuse, or not tringle. ute 2 right otuse not tringle 6 20 not tringle 11 lssif the tringle s ute, right, otuse, or not tringle. 1 Slide 36 () / 240 Slide 37 / 240 right otuse not tringle ute Tell whether the lengths 3, 6, nd 6 represent the sides of n ute, right, or otuse tringle. ute right lssif the tringle s ute, right, otuse, or not tringle. otuse 1 Slide 37 () / 240 right otuse ute 14 Tell whether the lengths represent the sides of n ute, right, or otuse tringle. ute tringle right tringle otuse tringle 13 Tell whether the lengths 3, 6, nd 6 represent the sides of n ute, right, or otuse tringle. Slide 3 / 240
11 Slide 3 () / 240 Slide 3 / Tell whether the lengths represent the sides of n ute, right, or otuse tringle. right tringle ute tringle Review If = +, then tringle is right. 2 otuse tringle 2 2 If 2 > 2 + 2, then tringle is otuse. If 2 < 2 + 2, then tringle is ute. Slide 40 / 240 Slide 41 / 240 There re mn proofs to the Pthgoren Theorem. How mn do ou know? Similrit in Right Tringles Slide 42 / 240 Theorem The ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd eh other. Slide 43 / 240 To prove this, for L 1 - Similr Right Tringles Teher Notes Return to the Tle of ontents Tringle similrit n e used to prove the Pthgoren Theorem. How? is the ltitude of ~ ~ Therefore, the ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd similr to eh other.
12 Slide 43 () / 240 Slide 44 / 240 See Similr Right Tringles L toomplete this tivit. The ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd eh other. 1. ut n inde rd long one of its digonls, forming two ongruent right tringles. 2. For one right tringle, drw n ltitude from the right ngle to the hpotenuse. ut long the ltitude to form two right tringles. 3. ompre the three tringles. Wht speil propert do the shre? Teher Notes Let's prove the Theorem. To prove this, for L 1 - Similr Right Tringles ~ Prove: ~ Resons Sttements Given is right tringle is right tringle is the ltitude of Given: [This ojet is pull t] Therefore, the ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd similr to eh other. is right ngle Given ef of ltitude ef of Perp Lines. 2 lines tht form rt ngle is right ngle ll rt ngles re Refleive Prop of ~ ~ is right ngle ef of Perp Lines ll rt ngles re Refleive Prop of ~ ~ ~ Slide 4 / 240 Mth up the ngles. e Helpful tip: If ou set, then ou n ssign ll the ngles vlue nd esil find the mthes d ~ ~ ~ ~ Slide 4 / 240 ltitude of rt tringle theorem. To prove the Pthgoren Theorem, use the proportions (ontinued). efinition of similr tringles. Given: Resons Sttements ~ Using the multiplition propert of equlit, multipl the eqution. Prove: (1) d e 30 is right tringle. is n ltitude. e 30 To prove the Pthgoren Theorem, use the proportions. euse the tringles re similr the orresponding sides re proportionl. Slide 47 / 240 Given: Lel the sides of tringle with the lower se letter of the opposite ngle ssign lengths to ll the segments. Let the lengths of the segments on the hpotenuse e d nd e. Trnsitive Prop of ~ Slide 46 / 240 Let's sketh the 3 tringle's seprtel, with the sme orienttion. ~ ~ simplif ltitude of rt tringle theorem. efinition of similr tringles. is right tringle. is n ltitude. Prove: Using the multiplition propert of equlit, multipl the eqution (2) simplif Resons Using the ddition propert of equlit, dd eqution (1) nd eqution (2) together. istriutive Propert Given Sustitution e d d Sttements Simplif
13 Slide 4 / 240 Slide 0 / 240 mple Find the length of the ltitude KI? H It me helpful to sketh the 3 tringle's seprtel, with the sme orienttion. H K 13 K K I H H 13 J J K I 13 I J K euse the tringles re similr the orresponding sides re proportionl. I J 13 = Slide 1 / 240 Tr this... Find the length of RS. 3 P Q Tr this... Find the length of RS. R 4 Slide 1 () / P Q Q S 4 Q P S R S P 4 3 P R R S S S R Q S 4 Q P R S Slide 2 / 240 Slide 2 () / Whih rtio is the rtio of orresponding sides? K J H H I 1 Whih rtio is the rtio of orresponding sides? I K J
14 Slide 3 / 240 Slide 3 () / Find KJ. 16 Find KJ. I I 24 J K 2 H 7 H 24 7 Set KJ = J K 2 Slide 4 / 240 The net two theorems re Geometri Men Theorems. Wht is men? n verge. Usull when we sk to find the men, we re sking for the rithmeti men. Wht is n rithmeti men? The sum of n vlues divided the numer of vlues (n). Wht is geometri men? The nth root of produt of n vlues. It is defined for onl positive numers (no negtive numers, no zero) Slide / 240 The geometri men of two positive numers nd is the positive numer tht stisfies = 2 = = Visull, the geometri men nswers this question: given retngle with sides nd, find the side of the squre whose re equls tht of the retngle. For more informtion on this link: rithmeti Men vs Geometri Men Slide 6 / 240 Slide 7 / Find the geometri men of 7 nd 6. Write the mple Find the geometri men of nd = (14) nswer is simplest rdil form. (onl the positive vlue) 2 = 1
15 Slide 7 () / 240 Slide / Find the geometri men of 7 nd 6. Write the 1 Find the geometri men of 3 nd 4. nswer is simplest rdil form. Students tpe their nswers here Slide () / 240 Slide / 240 orollr The ltitude drwn to the hpotenuse of right tringle divides the the hpotenuse into two segments. The ltitude is the geometri men of the two segments formed. 1 Find the geometri men of 3 nd 4. Students tpe their nswers here is the ltitude of Sine, ~ 2 = () Slide 60 / 240 Slide 60 () / 240 mple mple Find z. Find z. 6 6 z z
16 Slide 61 / 240 Slide 61 () / 240 mple mple Find z. Find z z z 1 Slide 62 / 240 Slide 62 () / 240 Tr this... Find. 2) 1) 2) 1 1 1) Tr this... Find. 1) = 2) = 6.7 Slide 63 / 240 Slide 63 () / Find Find.
17 Slide 64 / 240 Slide 64 () / Find. 20 Find Slide 6 / 240 Slide 66 / 240 = ~ ~ 4 R S S T U = Slide 66 () / 240 mple 4 R Find. Slide 67 / 240 mple T Find. Find. 4 F 6 U G is the ltitude of Sine, ~ ~ mple orollr If the ltitude drwn to the hpotenuse of right tringle, divides the hpotenuse into two segments. The length of eh leg of the originl tringle is the geometri men of the lengths of the hpotenuse nd the segment of the hpotenuse tht is djent to the leg.
18 Slide 67 () / 240 Find. 21 Is PR geometri men etween QR nd SR? 4 F P True 6 Flse Q G R S mple Slide 6 / 240 Slide 6 () / Is PR geometri men etween QR nd SR? True Flse Q Q Flse R S R S Flse 22 Is the geometri men orret? P P True Slide 6 / 240 Slide 6 () / Whih proportion is orret? P J Flse True S Q True R K L M 22 Is the geometri men orret? Slide 70 / 240
19 Slide 70 () / Whih proportion is orret? 24 Find. L 20 M 24 Find. Slide 72 / Find Slide 71 () / None of the ove K J Slide 71 / 240 Slide 72 () / Find None of the ove 2 Find. Slide 73 / 240
20 Slide 73 () / 240 Slide 74 / Find. Speil Right Tringles Return to the Tle of ontents Slide 7 / 240 Slide 76 / 240 In this setion ou will lern out the properties of the two speil right tringles Tringle Theorem tringle is n isoseles right tringle, where the hpotenuse is 2 times the length of the leg. 0o 60o 4o n ou prove this? 2 30o 0o 4o Slide 76 () / tringle is n isoseles right tringle, where the hpotenuse is 2 times the length of the leg. hpotenuse = leg( 2) 4o Slide 77 / Tringle Theorem n ou prove this? mple Find the length of the missing sides. Write the nswer in simplest rdil form. P 6 Q 4 o 4o 2 hpotenuse = leg( 2) 4o 4o 4o R
21 Slide 7 / 240 mple Find the length of the missing sides. Write the nswer in simplest rdil form. 6 the orollr to the P se ngles 4 Thm, PQ=QR. o =6 mple Find the length of the missing sides of the right tringle. hpotenuse = 2(leg) = 2(6) = 6 2 S Q T 1 Slide 77 () / 240 4o V R Slide 7 () / 240 Sine, STU is n isoseles rt tringle S hpotenuse = leg Tr this... Find the length of the missing sides. T ST=TV = There re 2 ws to solve. 1 mple Find the length of the missing sides of the right tringle. Slide 7 / 240 V Slide 7 () / Find the vlue of. ( 2)/2 Tr this... Find the length of the missing sides. Slide 0 / 240
22 Slide 0 () / 240 Slide 1 / Find the vlue of. ( 2)/2 2 2 ( 2)/2 27 Find the vlue of. Slide 1 () / 240 Slide 2 / Wht is the length of the hpotenuse of n isoseles right tringle, if the length of the legs is 2 inhes. 2 Find the vlue of. ( 2)/2 2 Slide 3 / Wht is the length of the hpotenuse of n isoseles right tringle, if the length of the legs is 2 inhes. hpotenuse = leg( ) 30 Wht is the length of eh leg of n isoseles, if the length of the hpotenuse is 20 m. Slide 2 () / 240
23 Slide 3 () / 240 Slide 4 / Tringle Theorem 30 Wht is the length of eh leg of n isoseles, if the length of the hpotenuse is 20 m.= leg( ) hpotenuse In right tringle, the hpotenuse is twie the length of the shorter leg nd the longer leg is 3 times the length of the shorter leg. Slide / = 3 G mple Find the length of the missing sides of the right tringle o o 30o =2 2 Slide 6 / o For right tringle, is perpendiulr isetor. let =, = 2 nd = 60o hpotenuse = 2(shorter leg) longer leg = 3(shorter leg) This n e proved using n equilterl tringle. 60 Slide 7 / 240 F Slide 7 () / 240 H 60o F GF is the longest side (hpotenuse) GH is the 2nd longest longerside leg = 3(shorter leg) HF < GH < GF = 3() = 3 G 30o 30o Rell tringle inequlit, the shortest side is opposite the smllest ngle nd the longest hpotenuse = 2(shorter leg) side is opposite the lrgest ngle. = 2() = HF is the shortest side G Rell tringle inequlit, the shortest side is opposite the smllest ngle nd the longest side is opposite the lrgest ngle. HF is the shortest side GF is the longest side (hpotenuse) GH is the 2nd longest side HF < GH < GF 60o H H 60o F
24 M 60 o mple Find the length of the missing sides of the right tringle. Slide () / 240 mple Find the length of the missing sides of the right tringle. Slide / o M is the shorter leg nd MT is the longer leg longer leg = 3(shorter leg) 30o 30o = 3() 3 3 = T T Slide / 240 M Slide 0 / 240 The ltitude (or height) divides the tringle into two 30o-60o-0o tringles. mple Find the re of the tringle. 14 ft h? 14 ft? The length of the shorter leg is 7 ft. The length of the longer leg is 7 3 ft. = (h) = 14(7 3) 4.7 squre ft Slide 1 / 240 Slide 1 () / 240 Tr this... Find the length of the missing sides of the right tringle. Tr this... Find the length of the missing sides of the right tringle o o 60 o 60 o
25 Slide 2 / 240 Slide 2 () / 240 Tr this... Find the re of the tringle. Tr this... Find the re of the tringle. ft ft 30o 30o short leg = 4. ft long leg = 4. ft Slide 3 / 240 Slide 3 () / Find the vlue of. 31 Find the vlue of o 7 30o (7 2)/ (7 2)/2 7 60o 7 30o 14 Slide 4 / 240 Slide 4 () / Find the vlue of. 32 Find the vlue of. (7 2)/ (7 2)/
26 Slide / o o (7 2)/ Find the vlue of o Find the vlue of. Slide () / 240 (7 2)/ o Slide 6 / 240 Slide 6 () / The hpotenuse of 30o -60o -0o tringle is 13 m. Wht is the length of the shorter leg? 34 The hpotenuse of 30o -60o -0o tringle is 13 m. Wht is the length of the shorter leg? shorter leg = 13/2 shorter leg = 6.m Slide 7 / 240 Slide 7 () / The length the longer leg of 30o -60o -0o tringle is 7 m. Wht is the length of the hpotenuse? 3 The length the longer leg of 30o -60o -0o tringle is 7 m. Wht is the length of the hpotenuse? shorter leg = hpotenuse = =
27 Slide / 240 Slide / 240 Rel World mple? 2. 30o The tringle formed the rmp is 30o-60o-0o right tringle. The length of the rmp is the hpotenuse. hpotenuse = 2(shorter leg) hpotenuse = 2(2.) hpotenuse = The rmp is feet long. The wheelhir rmp t our shool hs height of 2. feet nd rises t ngle of 30o. Wht is the length of the rmp? Slide 0 / 240 Slide 0 () / feet 4o? 36 skteorder onstruts rmp using plwood. The length of the plwood is 3 feet long nd flls t n ngle of 4. Wht is the height of the rmp? Round to the nerest hundredth. 36 skteorder onstruts rmp using plwood. The length of the plwood is 3 feet long nd flls t n ngle of 4. Wht is the height of the rmp? Round to the nerest hundredth. 4o 3 feet? Slide 1 / 240 Slide 1 () / feet? 4o 37 Wht is the length of the se of the rmp? Round to the nerest hundredth. 37 Wht is the length of the se of the rmp? Round to the nerest hundredth. 4o 3 feet?
28 Slide 2 / 240 Slide 2 () / The ield sign is shped like n equilterl tringle. Find the length of the ltitude. 3 The ield sign is shped like n equilterl tringle. Find the length of the ltitude. 20 inhes 20 inhes Slide 3 / 240 Slide 3 () / The ield sign is shped like n equilterl tringle. Find the re of the sign. 3 The ield sign is shped like n equilterl tringle. Find the re of the sign. 20 inhes 20 inhes Slide 4 / 240 Slide / 240 Right tringle trigonometr is the stud of the reltionships etween the sides nd ngles of right tringles. Trigonometri Rtios Return to the Tle of ontents
29 Slide 6 / 240 Slide 7 / 240 ngineers ver refull mesure the perpendiulr distne from tower window (points, or F) to the ground (points G, or ). Then the mesure the distne from the tower to points, or G. F ~ ~ FG ver sine the onstrution of the ell Tower in the 10's, it hs slowl tilted south nd is t risk of flling over. If the ngle of slnt ever fll's elow 3 degrees, it is fered the tower will ollpse. ngineers n mesure the ngle of slnt using n of the right tringles onstruted elow. WHY? Lening Tower of Pis, ell Tower in Pis, Itl G ngle of slnt Slide 7 () / 240 Slide / 240 ngineers n mesure the ngle of slnt using n of the right tringles onstruted elow. ngineers ver refull mesure the perpendiulr distne from tower window (points, or F) to the ground (points G, or ). Then the mesure the distne ~ from the tower to points, or G. ~ ~ FG Tringle Height se Rtio Height / se =0m =m 0/= =30m =3m 30/3= FG FG=20m G=2m 20/2= Let's lulte the rtio's of the height to the se for eh right tringle. F Notie tht ll of the rtios re the sme. WHY? WHY? G ngle of slnt Slide / 240 When the tringle is dilted (pull sle), how does the ngle hnge? Wht hppens to the slope rtio? Wht hppens to the rtio when the ngle inreses? Wht hppens to the rtio when the ngle dereses? lik for intertive wesite to investigte. The rtio of height/se is lso lled the slope rtio (rise/run) or tngent rtio. Slide 1 / 240 To lern right tringle trigonometr, first ou need to e le to identif the sides of right tringle. In right tringle, there re 2 ute ngles. In the tringle to the left, nd re the ute ngles. Lel the sides of tringle with the lower se letter of the opposite ngle.
30 Slide 111 / 240 Slide 1 / Wht is the side opposite to J? LK opp dj hp JL hp opp KJ K When is the referene ngle, the side opposite is. the side djent (or net to) is. nd the hpotenuse is. Slide 1 () / 240 Slide 113 / Wht is the side opposite to J? JL 41 Wht is the hpotenuse of the tringle? JL L J L J LK LK KJ L J KJ K K Let's look t, when is the referene ngle, the side opposite is. the side djent (or net to) is. nd the hpotenuse is. dj Slide 113 () / 240 Slide 114 / Wht is the hpotenuse of the tringle? KJ LK L J J L LK KJ K JL K JL 42 Wht is the side djent to J?
31 Slide 114 () / 240 Slide 11 / Wht is the side opposite K? L J JL L J LK LK KJ JL K 42 Wht is the side djent to J? K KJ Slide 11 () / 240 Slide 116 / Wht is the side opposite K? 44 Wht is the side djent to K? KJ JL LK K JL LK L J L J K KJ Slide 116 () / Wht is the side djent to K? Slide 117 / 240 L J trigonometri rtio is the rtio of the two sides of right tringle. KJ JL LK Trigonometri Rtios K There re 3 rtios for eh ute ngle of right tringle. The rtios re lled sine, osine, nd tngent (revited sin, os, nd tn).
32 Slide 11 / 240 Slide 11 / 240 The 3 Trigonometri Rtios opposite side hpotenuse osθ = djent side hpotenuse opposite side djent side tnθ = This spells... SOHHTO θ or whih is pneumoni to help ou rememer the sides of right tringle (ou'll need to rememer the spelling). Slide 0 / 240 Slide 1 / 240 mple Find the sin F, os F, nd tn F. 6 Find the sin, os, nd tn. Sine F is our referene ngle, lel the sides of the tringle opposite, djent nd hpotenuse. Use the pneumoni to find the trig rtios. lws redue opp 6 dj sinf hp F opp = hp frtions to lowest terms. 3 6 = = tn F = frtions to lowest terms. dj 3 4 opp Slide 2 / 240 F os = dj = 6 = 3 hp tn = opp dj 4 = =3 6 4 Wht is the sin R? 20/2 21/20 21/20 20/2 20/21 sin hp opp 4 = hp = = Slide 2 () / Wht is the sin R? 21/2 F Sine is our referene ngle, lel the sides of the tringle opposite, djent nd hpotenuse. Use the pneumoni to find the trig rtios. lws redue 6 osf = dj = = 4 hp = opp = 6 dj 6 F mple 21/2 sinθ = lik for SOHHTO song on outue.om "Gettin' Trigg Wit It". 20/21
33 Slide 3 / 240 Slide 3 () / Wht is the osr? 20/2 21/20 21/20 20/2 21/2 20/21 21/2 20/21 46 Wht is the osr? Slide 4 / 240 Slide 4 () / Wht is the tnr? 20/21 21/20 21/20 20/21 20/2 21/2 47 Wht is the tnr? 20/2 21/2 Slide / 240 Slide () / Wht is the sinq? 4 Wht is the sinq? 21/2 2/20 21/20 21/2 21/20 20/2 20/2 2/20
34 Slide 6 / 240 Slide 6 () / Wht is the osq? 20/2 21/20 20/2 21/20 21/2 21/2 2/21 2/21 Slide 7 / Wht is the osq? Slide 7 () / Wht is the tnq? 0 Wht is the tnq? 20/2 21/20 21/2 20/21 21/20 21/2 20/2 20/21 Slide / 240 Slide / 240 The ngle of slnt of the Tower of Pis is vlute sin 60. Round to the nerest ten thousndth. To find the trigonometri rtio of n ngle, use lultor or trig tle. Find the following: sin 4.3 =.1 os 4.3 =.03 tn 4.3 = F ngle of slnt hek tht our lultor is set for degrees (not rdins) nd round our nswer to the ten thousndth ple (4 deiml ples).
35 Slide () / vlute sin 60. Round to the nerest ten thousndth Slide 130 () / vlute os 60. Round to the nerest ten thousndth Slide 131 / vlute tn 60. Round to the nerest ten thousndth vlute os 60. Round to the nerest ten thousndth Slide 130 / Slide 131 () / vlute tn 60. Round to the nerest ten thousndth Trig tles were used erl mthemtiins nd stronomers to lulte distnes tht the were unle to mesure diretl. Tod, lultors re usull used Slide 132 /
36 Slide 133 / 240 Slide 134 / 240 How do ou find n unknown side mesure in right tringle when ou re given n ute ngle nd one side? mple Find the trig eqution tht will find. You need to identif the orret trig funtion tht will find the missing side. Use SOHHTO to help. Using, I m looking for o nd I hve, so the rtio is o/ whih is tngent. now ou n solve for, the missing side. opp 30o dj Slide 134 () / 240 dj 30o 30o opp 30 o Slide 13 / 240 mple Find the trig eqution tht will find. mple Find the trig eqution tht will find. is our ngle of referene. Lel the given nd unknown sides of our tringle opp, dj, or hp. Identif the trig funtion tht uses, the unknown side nd the given side. Slide 13 () / 240 mple Find the trig eqution tht will find. mple Find the trig eqution tht will find. 30o 30o dj 30o hp Slide 136 / 240
37 Slide 136 () / 240 Slide 137 / 240 mple Find the trig eqution tht will find. 4 Using, whih is the orret trig eqution needed to solve for. hp o os40 = / 40 tn40 = / opp Slide 13 / Using, whih is the orret trig eqution needed to solve for. Using, whih is the orret trig eqution needed to solve for. tn40 = / sin0 = / 0o 4 os0 = / sin0 = / 0o Slide 13 () / Using J, whih is the orret trig eqution needed to solve for. sin0 = / tn32 = /11 sin0 = / Slide 13 / 240 Using, whih is the orret trig eqution needed to solve for. tn0 = / tn0 = / sin40 = / os0 = / Slide 137 () / 240 os40 = / sin40 = / sin40 = / 30o sin40 = / 0o 11 os32 = /11 tn32 = 11/ sin32 = 11/ K J 32o L 30 o
38 Slide 140 / 240 tn32 = /11 os32 = /11 tn32 = 11/ 6 Using J, whih is the orret trig eqution needed to solve for. K sin32 = 11/ os = /11 tn = 11/ 32o o tn = /11 11 J 7 Using K, whih is the orret trig eqution needed to solve for. L sin = 11/ J K 11 Slide 13 () / 240 L Slide 140 () / 240 Slide 141 / Using K, whih is the orret trig eqution needed to solve for. os = /11 tn = 11/ sin = 11/ o tn = /11 J Finding the Missing Side of Right Tringle K Now, ou n solve for, the missing side. Round our nswer to the nerest tenth. Using our lultor, find the tn 4.3 Round our nswer to 4 deiml ples. 11 opp You n rewrite.017 with denomintor of 1 nd use the ross produt propert or multipl oth sides of the eqution using the multiplition propert of equlit (see net slide). L dj Slide 142 / 240 Slide 143 / 240 mple Find. Round our nswer to the nerest hundredth. Now, ou n solve for, the missing side. Round our nswer to the nerest tenth. G opp 2o Multipl oth sides of the eqution using the multiplition propert of equlit. M dj Finding the Missing Side of Right Tringle
39 Slide 143 () / 240 Slide 144 / 240 2o G sin G = M GM sin2 = G ().4226 = mple Find. Round our nswer to the nerest hundredth. () mple Find. Round our nswer to the nerest hundredth..07 o 6 M M Slide 144 () / 240 Slide 14 / 240 mple Find. Round our nswer to the nerest hundredth. os M = M GM os 6 = ().4226 = o 6 G mple Find. Round our nswer to the nerest hundredth. 20o ().07 M Slide 146 / o mple Find. Round our nswer to the nerest hundredth. tn = tn 20 = ().3640 = Find the length of LM. Round our nswer to the nerest tenth. P ().3640 = L 6o M Slide 14 () / 240
40 Find the length of LP. Round our nswer to the nerest tenth. P Find the length of LM. Round our nswer to the nerest tenth. Slide 147 / 240 L 6o Find the length of LP. Round our nswer to the nerest tenth. L M Slide 147 () / 240 P 6o M Slide 14 / 240 plin nd use the reltionship etween the sine nd osine of omplementr ngles. P L 6o Slide 14 / 240 M Slide / 240 To find the mesure of... Find the mesure of? The sum of the interior ngles of n tringle is equl to degrees. nd re omplementr ngles. omplementr ngles re two ngles whose sum of their mesures is 0 degrees. The ute ngles of right tringle re lws omplementr. Slide 146 () / 240
41 Slide 11 / 240 Slide 11 () / For right tringle, wht is the mesure of? 60 For right tringle, wht is the mesure of? 60 degrees nnot e determined 0 degrees 60 degrees 0 degrees nnot e determined 30 o Slide 12 / 240 [Thisoojet is pull t] 30 Slide 12 () / 240, find the omplementr ngle? 61 If the 20 degrees 20 degrees, find the omplementr ngle? 70 degrees 160 degrees none of the ove 70 degrees 160 degrees 61 If the 30 degrees 30 degrees none of the ove Slide 13 / 240 Slide 14 / 240 Let's ompre the sine nd osine of the ute ngles of right tringle. In right tringle, the ute ngles re omplementr. m + m = = 0 First, find the mesure of LP using the sine funtion. Then, find the mesure of LP using the osine funtion. sine funtion osine funtion 4 sin = 4/ sin 3.1 =.77 os = 4/ os 36. =.77 sin = os 3.1 sin 3.1 = os The sine of n ngle is equl to the osine of its omplement. os = 3/ os 3.1 =.6004 sin = 3/ sin 36. =.6004 os = sin os 3.1 = sin 36. The osine of n ngle is equl to the sine of its omplement. L 6o Sine nd osine re lled o-funtions of eh other. o-funtions of omplementr ngles re equl. P 22o M
42 Slide 1 / 240 Slide 1 () / Given tht sin =.1736, write the osine of 62 Given tht sin =.1736, write the osine of sin =.1736 sin =.1736 sin 0 =.4 sin 0 =.4 os =.4 os =.4 os 0 =.1736 os 0 =.1736 Slide 16 / 240 omplementr ngle. omplementr ngle. Slide 16 () / Given tht os 0 =.642, write the sine of 63 Given tht os 0 =.642, write the sine of sin 0 =.7660 sin 0 =.7660 sin 40 =.642 os 0 =.642 os 0 =.642 os 40 =.7660 os 40 =.7660 Slide 17 / 240 omplementr ngle. omplementr ngle. sin 40 = Given tht os 6 =.4226, write the sine of sin 2 =.4226 sin 2 =.4226 os 6 =.4226 os 2 =.063 sin 6 =.063 os 6 =.4226 omplementr ngle. omplementr ngle. sin 6 =.063 Slide 17 () / Given tht os 6 =.4226, write the sine of os 2 =.063
43 Slide 1 () / Wht n ou onlude out the sine nd osine of 4 degrees? Students tpe their nswers here 6 Wht n ou onlude out the sine nd osine of 4 degrees? Students tpe their nswers here Slide 1 / 240 sin 4 = os 4 Slide 1 / 240 Slide 160 / 240 To solve right tringle mens to find ll 6 vlues in tringle. The lengths of ll 3 sides nd the mesures of ll 3 ngles. Solving Right Tringles Slide 161 / 240 Let's solve right tringle given the length of one side nd the mesure of one ute n gle (S). You need to find the 3 missing prts. Slide 162 / 240 First, let's find the mesure of o z 64o z Return to the Tle of ontents
44 Slide 162 () / 240 Slide 163 / 240 First, let's find the mesure of. Then, let's find the mesure of. 26 m< + m< = 0o o 64o + m< = 0 64o z o m< = 26 64o z 1 1 Slide 163 () / 240 Slide 164 / 240 Then, let's find the mesure of. Then, let's find the mesure of o sin64 = 64 z 1. = o 13.4 z 1 Slide 164 () / 240 Slide 16 / 240 Then, let's find the mesure of. Tr this... Find the missing prts of the tringle. 26 R = 2 64o 2 = 1 z 2 z2 + (13.4) z = 22 z2 = 43.2 z o 1
45 Slide 16 () / 240 Slide 166 / 240 Tr this... Find the missing prts of the tringle. Let's solve right tringle given the length of two sides (SS). R 11 37o R m R = 3o z 1 Slide 167 / 240 Slide 167 () / 240 First, find the length of sine we know how to do tht. ut, how do ou find the mesure of nd? z 1 First, find the length of sine we know how to do tht. ut, how do ou find the mesure of nd? = 2 z2 + 2 = 12 z2 + 1 = 22 z z21 = 144 z = = Slide 16 / 240 Slide 16 / 240 The 3 Inverse Trigonometri Rtios You will need to use the inverse trig funtions. If sinθ =, θ = sin-1 If osθ =, θ = os-1 If tnθ =, θ = tn-1 Pronouned inverse sine, inverse osine, nd inverse tngent. θ θ = sin-1( opposite side ) hpotenuse θ = tn-1( opposite side ) θ = os-1( djent side ) djent side hpotenuse Use the inverse trig funtion to find the unknown ngle mesure when ou know the length of 2 sides. With the sine, osine nd tngent trig funtions, if ou know the ngle θ nd the mesure of one leg, then ou n find the mesure of leg of tringle. With the inverse sine, inverse osine nd inverse tngent trig funtions, if ou know the mesures of 2 legs of tringle, ou n find the mesure of the ngle. Rememer: θ
46 Slide 170 / 240 Slide 170 () / Find sin Round the ngle mesure to the nerest hundredth. 66 Find sin Round the ngle mesure to the nerest hundredth. θ = 3.13 Slide 171 / 240 Slide 171 () / Find tn Round the ngle mesure to the nerest hundredth. 67 Find tn Round the ngle mesure to the nerest hundredth. θ = 66.0 Slide 172 / 240 Slide 172 () / Find os Round the ngle mesure to the nerest hundredth. 6 Find os Round the ngle mesure to the nerest hundredth. θ = 63.26
47 Slide 173 / 240 Slide 173 () / 240 To find n unknown ngle mesure in right tringle, To find n unknown ngle mesure in right tringle, You need to identif the orret trig funtiontht will find the missing vlue. Use SOHHTO to help. You need to identif the orret trig funtiontht will find the missing vlue. Use SOHHTO to help. Using osine. dj θ 1 hp, I hve nd h, so the rtio is /h whih is Using osine dj θ 1 1 now ou n solve for, the[this missing ojet is pullngle t] using the inverse trig funtion. How re ou going to lulte the mesure of? How re ou going to lulte the mesure of? Slide 174 () / 240 mple Find the trig eqution tht will find θ. θ 7 θ 7 θ dj 7 opp Slide 17 / 240 Slide 17 () / 240 mple Find the trig eqution tht will find θ. mple Find the trig eqution tht will find θ. θ θ hp now ou n solve for, the missing ngle using the inverse trig funtion. mple Find the trig eqution tht will find θ., I hve nd h, so the rtio is /h whih is Slide 174 / 240 is our ngle of referene. Lel the two given sides of our tringle opp, dj, or hp. Identif the trig funtion tht uses, nd the two sides. is our ngle of referene. Lel the two given sides of our tringle opp, dj, or hp. Identif the trig funtion tht uses, nd the two sides. dj θ hp
48 Slide 176 / 240 Slide 176 () / 240 mple Find the trig eqution tht will find θ. mple Find the trig eqution tht will find θ. θ θ θ hp opp Slide 177 / 240 Slide 177 () / Whih is the orret trig eqution to solve for 6 Whih is the orret trig eqution to solve for 7 7 Slide 17 / 240 Slide 17 () / Whih is the orret trig eqution to solve for 70 Whih is the orret trig eqution to solve for
49 Slide 17 / 240 Slide 17 () / Whih is the orret trig eqution to solve for K 11 J K 71 Whih is the orret trig eqution to solve for L 11 J L Slide / 240 Tr this... Solve the right tringle. Round our nswers to the nerest hundredth. R 24 Q S Q R 24 7 Tr this... Solve the right tringle. Round our nswers to the nerest hundredth. Slide () / 240 S QS = 2 m Q = 73.74o m S = 16.26o Slide 11 / 240 Slide 11 () / Find. Use the Pthgoren Theorem 72 Find. 7
50 Slide 12 / 240 Slide 12 () / Find m. 73 Find m. Slide 13 / 240 Slide 13 () / Find the m. 74 Find the m. From efore, Slide 14 / 240 Slide 14 () / Find the m G. 7 Find the m G. o G 20 o 20 1 L L Use inverse tngent 1 G
51 Slide 1 / 240 Slide 1 () / 240 o 20 o Find L. L 76 Find L. L 1 G G 1 Slide 16 / 240 Slide 16 () / Find the m P. 77 Find the m P. P 4.1o P 4.1o 33.6o 41.1o 6.31o o 41.1o 6.31o N 1 N Slide 17 / Find RT. S Slide 17 () / Find RT. T o R T S 40o R
52 Slide 1 / 240 Slide 1 / 240 How n ou use trigonometri rtios to solve word prolems involving ngles of elevtion nd depression? ngle of levtion nd epression Return to the Tle of ontents Slide / 240 Slide 11 / 240 When ou look up t n ojet, the ngle our line of sight mkes with line drwn horizontll is the ngle of elevtion. When ou look down t n ojet, the ngle our line of sight mkes with line drwn horizontll is the ngle of depression. The ngle of elevtion nd the ngle of depression re oth mesured reltive to prllel horizontl lines, the re equl in meure. Slide 13 / How n ou desrie the ngle reltionship etween the ngle of elevtion nd the ngle of depression? orresponding ngles lternte interior ngles lternte eterior ngles none of the ove Slide 12 / 240
53 Slide 13 () / How n ou desrie the ngle reltionship etween the ngle of elevtion nd the ngle of depression? orresponding ngles lternte interior ngles mple m is fling kite t n ngle of o. The kite's string is 1 feet long nd m's rm is 3 feet off the ground. lternte eterior ngles none of the ove Slide 14 / f t o How high is the kite off the ground? ee 3 feet Slide 1 / f t Slide 16 / 240 sinθ = 1 sin = 1.40 = o 1 mple You re stnding on mountin tht is 306 feet high. You look down t our mpsite t ngle of 30o. If ou re 6 feet tll, how fr is the se of the mountin from the mpsite? = 134 Now, we must dd in m's rm height = o 6 ft The kite is out 137 feet off the ground. 306 ft Slide 17 / 240 Slide 1 / 240 Tr this... 3 ft 30o.774 = = 3,200 ft The mpsite is out,200 ft from the se of the mountin. You re looking t the top of tree. The ngle ofelevtion is o. The distne from the top of the tree to our position (line of sight) is 4 feet. If ou re. feet tll, how fr re ou from the se of the tree? tn30 =
54 Slide 1 () / 240 You re looking t the top of tree. The ngle ofelevtion is o. The distne from the top of the tree toftour position (line of sight) is 4 feet. If ou re. feet tll,4how fr re ou from the se of the tree? o os = 0 When ou look down t n ojet, the ngle our line of sight mkes with line drwn horizontll is the ngle of. elevtion Tr this... Slide 1 / 240 depression 4 = 4.1 You re pproimtel 4 ft ojetof is the pull t] from the[this se tree. Slide 1 () / 240 depression 1 Ktherine looks down out of the rown of the sttue of liert to n inoming ferr out 34 feet. The distne from rown to the ground is out 20 feet. Wht is the ngle of depression? elevtion 0 When ou look down t n ojet, the ngle our line of sight mkes with line drwn horizontll is the ngle of. Slide 200 / Ktherine looks down out of the rown of the sttue of liert to n inoming ferr out 34 feet. The rown distne from rown to the ground is out 20 feet. Wht is the ngle of depression? 34 ft 20 ft ferr The ngle of[this depression out ojet is is pull t]46 degrees. Slide 201 / Wht is the distne from the ferr to the se of the sttue? Slide 200 () / 240
55 Slide 201 () / 240 Slide 202 / Wht is the distne from therown ferr to the se of the sttue? 34 ft 20 ft se of the sttue Lw of Sines nd Lw of osines ferr Return to the Tle of ontents The ferr is out 23 feet w from the sttue. Slide 203 / 240 Slide 204 / 240 How n ou solve non-right tringle? How n ou find the side lengths nd ngle mesures of non-right tringles? The Lw of Sines nd Lw of osines n e used to solve n tringle. You n use the Lw of Sines when ou re given 1. Two ngle mesures nd n side length (S or S) 2. Two side lengths nd the mesure of non-inluded ngle (SS) when the ngle is right ngle or n otuse ngle. The Lw of Sines hs prolem deling with SS when the ngle is ute. There n e zero, one or two solutions. To use the Lw of Sines, 2 ngles nd 1 side must e given. Given: hs sides of length,, nd Prove: sin = sin = sin Slide 206 / 240 h Prove the Lw of Sines (ontinued) Given: Sttements sin = sin = sin Slide 20 / 240 If hs sides of length,, nd, then sin = sin = sin If hs sides of length,, nd, then You n use the Lw of osines when ou re given 3. Three side lengths (SSS) 4. Two side lengths nd the mesure of n inluded ngle (SS) lik on: Khn dem Video "More On Wh SS Is Not Postulte" for more info. Let's prove the Lw of Sines Lw of Sines Resons with side lengths,, nd Given ef of ltitude rw n ltitude from to side Let h e the length of the ltitude ef of sine hs sides of length,, nd Prove: sin = sin = sin h g Sttements Resons ef of ltitude rw n ltitude from to side Let g e the length of the ltitude ef of sine Multipl. Mult Prop of =. Multipl. Mult Prop of =. Multipl. Mult Prop of =. Multipl. Mult Prop of =. Sustitution Prop of = Sustitution Prop of = ivide. ivision Prop of = ivide. ivision Prop of = Sustitution Prop of =
56 Slide 207 / 240 sin = sin = sin Selet the rtios sed on the given informtion. 70o 6o Given: m, m nd (side ) (S) Selet the rtios sed on the given informtion. Whih rtios must e used first? Wh? sin = sin = sin Use the Lw of Sines to solve the tringle. 70o 6o Given: m, m nd (side ) (S) Use the Lw of Sines to solve the tringle. Slide 207 () / 240 sinmust = sin Whih rtios e used first? Wh? There re 4 numers in proportion. If ou know 3 of the numers ou n find the 4th. Slide 20 / o First we n find the length side. 6o sin = sin sin70 = sin6 70o 6o sin = sin.37 =.063 sin70 = sin6.063 =.37 First we n find the length side. Slide 20 () / Slide 20 / o 6o Tringle Sum Theorem m + m + m = o =.37 efore we find the length of side, we find the m. 70o 6o =.37 Tringle Sum Theorem m + m + m = o efore we find the length of side, we find the m. Slide 20 () / 240 m + 70o + 6o = o m + 13o = o m = 4o
57 Slide 2 / 240 Slide 2 () / 240 6o 70o Now we find the length side. =4o =.37 sin = sin sin = sin 70o 6o Now we find the length side. =4o =.37 sin4 = sin = = Slide 211 / 240 Slide 211 () / 240 Tr this... Use the Lw of Sines to find the length of side (S). o 2o Sine the length of the side opposite < is given, find the m< first. hint Sine the length of the side opposite < is given, find the m< first. hint o 2o Tr this... Use the Lw of Sines to find the length of side (S)..1 Slide 2 / 240 Slide 2 () / 240 mple... Find the length of side (SS with n otuse ngle). 2. 1o mple... Find the length of side (SS with n otuse ngle). m< = sin = sin m<=.1 sin = sin1 sin = sin o sin20.0 = sin.1 2. sin = = = 6. sin =.3436 =sin-1(.3436) =20.0
58 Slide 213 / 240 Slide 213 () / Find the m. 3 Find the m. 31 o 1o 2o 1o 2o 70o 31 o 70o 1o 2o 2o 1o Slide 214 / 240 Slide 214 () / Whih rtio must e used to find the length of or? 70o sin sin sin Slide 21 / 240 sin sin Slide 21 () / 240 Wht is the length of? Wht is the length of? 1o 1o 70o 70o 1o sin sin 1o 70o sin 4 Whih rtio must e used to find the length of or?
59 Slide 216 / 240 Slide 216 () / Wht is the length of? 6 Wht is the length of? 70o 70o 1o 1o Slide 217 / 240 Slide 21 / 240 Lw of osines If hs sides of length,, nd, then: Given: hs sides of length,, nd Prove: (similr resoning shows tht To use the Lw osines, ou must e given the length of 3 sides (SSS) or the length of 2 sides nd the mesure of the inluded ngle (SS). ) If hs sides of length,, nd, then Let's prove the Lw of osines Sttements h - Resons with side lengths,, nd Given ef of ltitude rw n ltitude from to side. Let h e the length of the lt. Let e the length of. Then (-) is the length of. Segment ddition Postulte In efinition of osine, os = / (1) =(os) Multipl. Mult Prop of =. (2) In Pthgoren Theorem In,, Pthgoren Theorem Simplif Sustitution, eqution (2) ssoitive Prop of ddition Sustitution, eqution (1) Slide 21 / 240 Slide 220 / 240 mple Use the Lw of osines to solve the right tringle. =16 =27 =16 is opposite < is opposite < is opposite < =27 =23 The formul ou hoose depends on whih ngle ou re solving for first. To find the m, =23 2 = (os) 162 = (23)(27)(os) 26 = (os) 26 = - 42(os) -02 = -42(os).06 = os = os-1(.06) m 36.22o
60 Slide 221 / 240 Slide 222 / 240 =27 =16 = To find the m, 2 = (os) 232 = (16)(27)(os) 2 = (os) 7 = - 64(os) -406 = -64(os).46 = os =os-1(.46) m 61.7o =23 = =23 To find the m, or Use the Tringle Sum Theorem. Using 2 different methods, the nswers re slightl different euse of rounding. Slide 222 () / 240 Tr this... Use the Lw of osines to find the m< (SSS) =27 =23 m + m + m = o 36.22o o + m = o.1o + m = o Use the Tringle Sum Theorem. m 1.1o To find the m, =16 Slide 223 / 240 Slide 223 () / = (os) 72 = ()(6)(os) 4 = (os) 4 = 61-60os - = -60os.2 = os m< 7.46o 7 In the tringle the length of is Tr this... Use the Lw of osines to find the m< (SSS). Slide 224 / 240
61 Slide 224 () / 240 Slide 22 / In the tringle the length of is In the tringle the length of is... Slide 22 () / 240 Slide 226 / 240 In the tringle the length of is = (os) 1 2 = (os) 2 = (os) 2 = (os) Whih formul would ou use to find the m<? Slide 226 () / 240 Whih formul would ou use to find the m<? 1 2 = (os) 2 = (os) 0 Wht is the m? 2 = (os) 2 = (os) Slide 227 / 240
62 Slide 227 () / Wht is the m? Slide 22 / Wht is the m? 1 2 = (os) 2 = ()(1)(os) 64 = (os) 64 = os -242 = -270os.63 = os m 26.32o 1 Slide 22 () / Wht is the mesure of (S)? 1 Wht is the m? Students tpe their nswers here 1 2 = (os) 12 = ()()(os) 22 = (os) 22 = os 0 = -144os -.6 = os =os-1(-.6) m 3.7o 4 0 Slide 22 / 240 Slide 22 () / Wht is the mesure of (S)? 3 The Lw of Sines nd osines is used to solve... or 2 = (os) 2 = (6.23)(4)(os) 64 = (os) 4 = os -. = -4.4os.1166 = os m 3.3o right tringles ute tringles otuse tringles ll tringles Students tpe their nswers here 2 = (os0) 2 = ()(4)(.642) 4 2 = = = 6.23 Slide 230 / 240
63 Slide 230 () / 240 Slide 231 / The Lw of Sines nd osines is used to solve... right tringles ute tringles ll tringles re of n Olique Tringle otuse tringles Return to the Tle of ontents Slide 232 / 240 Slide 233 / 240 o ou rememer this? Previousl, we found the re of tringle when we were given 3 sides. Find the re of the tringle. 13 feet = 1 2 h is the se of the tringle =. h is the ltitude (or height). It is the perpendiulr isetor of the se in n isoseles tringle. Find h, using the pthgoren theorem - 13 feet 13 feet 13 feet h feet feet Slide 234 / 240 feet Slide 23 / 240 Wht formul n ou use to find the re of tringle if ou know the length of two sides nd the mesure of n inluded ngle (SS)? Sine = 1 h 2 nd =, we need to find h. 13 feet Find the re of the tringle. h feet feet 67.3 feet
64 Slide 236 / 240 Slide 237 / 240 Let's derive the formul for n olique tringle. hs sides of length,, nd. ltitude h. Prove: h Resons Sttements with side lengths,, nd Given rw n ltitude from to side ef of ltitude Let h e the length of the ltitude 4 Whih of the following epressions n e used to find the re of the tringle elow? Selet ll tht ppl. ef of sine F Given: Multipl. Mult Prop of =. efinition. Formul for the re of tringle. Sustitution Prop of = ommuttive Prop of Multiplition Slide 237 () / 240 Find the re of the tringle to the nerest tenth. Students tpe their nswers here 4 Whih of the following epressions n e used to find the re of the tringle elow? Selet ll tht ppl. Slide 23 / 240,, F Slide 23 () / 240 Find the re of the tringle to the nerest tenth. 6 Find the re of the tringle to the nerest tenth. Students tpe their nswers here Students tpe their nswers here Slide 23 / 240
65 Slide 23 () / Find the re of the tringle to the nerest tenth. 7 Find the re of the tringle to the nerest tenth. Students tpe their nswers here Students tpe their nswers here Slide 240 / 240 Slide 240 () / Find the re of the tringle to the nerest tenth. Students tpe their nswers here or
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