Chp. 3_4 Trigonometry.noteook Wrm Up Determine the mesure of the vrile in ech of the following digrms: x + 2 x x 5 x + 3 Pythgoren Theorem - is fundmentl reltionship mongst the sides on RIGHT tringle. c (hypotenuse) Pythgoren Triples C OPTIONS... #1. Finding the unknown hypotenuse: ex: Verifying Pythgoren Triple... 14 11 LS RS LS RS x #2. Finding n unknown side ex: 9 15 y Trigonometric Tle: Trigonometric Rtios *** Must hve clcultor in DEGREE mode *** Primry Trigonometric Rtios HYPOTENUSE OPPOSITE Reference ngle θ DJCENT Memory id: "SOH CH TO" Reciprocl Trigonometric Rtios Notice tht these rtios re ech the reciprocl of one of the primry trig rtios Summry Primry Rtios Reciprocl Rtios 1
Chp. 3_4 Trigonometry.noteook Reciprocl rtios re not found on clcultor...we must lern how to use the reciprocl function on our clcultor. EXMPLE - Finding n unknown side 12.8 cm 44o Reciprocl Functions 8 23o x or y Inverse Trigonometric Functions (rc Trig Functions) Trigonometric Functions EXMPLE - Finding n unknown ngle 18.1 m Evlute ech of the following: 14.3 m θ EXMPLE - Solve the tringle (find LL sides nd ngles) M L 15 mm I 35o 10.25 39 mm G or HOMEWORK... Worksheet Primry Trig Rtios.doc Wrm Up 1. Evlute ech of the following: () csc = 1.1924 = () sec 168o = 2. Solve the following tringles: () ΔRST, given tht S = 900, r = 12 cm nd t = 25 cm. () ΔMVH, given tht M = 900, H = 14o nd m = 44 cm. pplictions of Right ngle Trigonometry NGLE OF ELEVTION/DEPRESSION Exmple 1: Exmple 2: 2
Chp. 3_4 Trigonometry.noteook Exmple 3 Exmple 4 n ntenn is on the top of the CN Tower in Toronto. From point 2400 m wy, the ngles of elevtion to the top nd ottom of the ntenn re 12.1 o nd 9.9 o respectively. How tll is the ntenn? Wrm Up 2. new ridge is to e uilt cross gorge which is known to e 15 m wide. support pier is to e uilt t the deepest point of the gorge. If the ngles of depression to tht point re 39 o nd 58 o from the two ends of the ridge, wht must the height of this support pier e? 1. surveyor who wishes to know the width of river sights tree on the opposite nk s ering N 72 o E. He then wlks 46 m due est long the nk of the river until he is directly cross the river from the tree. How wide is the river? 3
Chp. 3_4 Trigonometry.noteook Exmples... REVIEW of... pplictions of Trig Rtios Exmples... REVIEW of... pplictions of Trig Rtios #1. The Goodyer limp is 565 m ove the ground during Super owl gme. The ngle of depression of the north gol line from the limp is 58.5 o. How fr is the oserver in the limp from the gol line? #1. The Goodyer limp is 565 m ove the ground during Super owl gme. The ngle of depression of the north gol line from the limp is 58.5 o. How fr is the oserver in the limp from the gol line? #2. n ntenn is on the top of the CN Tower in Toronto. From point 2400 m wy, the ngles of elevtion to the top nd ottom of the ntenn re 12.1 o nd 9.9 o respectively. How tll is the ntenn? #2. n ntenn is on the top of the CN Tower in Toronto. From point 2400 m wy, the ngles of elevtion to the top nd ottom of the ntenn re 12.1 o nd 9.9 o respectively. How tll is the ntenn? HOMEWORK... (Solving Right Tringles / Word Prolems) Worksheet pplictions of Primry Trig.doc Wrm Up 70 ft. 4
128 o 33 o Chp. 3_4 Trigonometry.noteook Lw of Sines ** Used when the tringle does not contin 90 o ngle (Olique Tringle) ** In order to use you must e given 1)n ngle nd n opposite side ND 2) ny other side or ngle EXMPLE #1 - Finding side. x 35.4 cm c Lower cse letters ",,c" represent side lengths Upper cse letters ",,C" represent ngle mesures EXMPLE #2 - Finding n ngle. θ 32 o 15 22 Lw of Sines "when looking for side" "when looking for n ngle" EXMPLE #3 - Solve the tringle. 14 25 o EXMPLE #4 - ppliction Suppose tht Mr. Wtters ws plying stright pr 3 golf hole tht ws 120 m long. He hits one of his regulr old slices tht ends up 40 o off line nd is still 82 m from the hole. () How fr did his tee shot trvel? () If he somehow mirculously hits his next shot onto the green, wht ws the ngle etween the pth of his first shot nd the pth followed y the second shot? Homework... Worksheet Lw of Sines.doc 24 Left Side... Right Side... #1 6 #1 4 WRM-UP... Lw of Cosines Derivtion of the lw of cosines... sk yourself... 1. Wht m I given? 2. Wht m I trying to find? c h m n C EXMPLE... On spce flight, stronnt Neil rmstrong reports tht the ngle formed y his lines of sight to the erth nd to the moon ws 58 0. t the sme time, the oserver on the erth reports tht the ngle formed y her lines of sight to the spceship nd to the moon is 74 0. If the moon is 382 000 km from the erth, how fr is the spceship from the trcking sttion? Finding n unknown side... 2 sides nd contined ngle (SS) c C Finding n unknown ngle... 3 known sides (SSS) c C 5
Chp. 3_4 Trigonometry.noteook EXMPLE: Finding n unknown side. 22 28 o 15 x ppliction Questions - Lw of Cosines sk yourself... 1. Wht m I given? 2. Wht m I trying to find? EXMPLE: Finding n unknown ngle. 2 m 5 m 6 m θ C EXMPLE... hockey net is 1.83m wide. plyer shoots from point where the puck is 13m from one gol post nd 11.5m from the other. Within wht ngle must he mke his shot to score? Exmple #2: From T, golfer ims ll towrds the hole t H which is 100m wy. ut the ll ctully sliced in direction 30 0 off course nd lnds t M, 60m wy. If the next shot is hit 50 m towrds the hole, will the ll go in the hole? sketch digrm M 60 m T 30 o 100m H REVIEW Wht formul do I use? sk yourself... Is it right tringle? If Yes, then... Homework... Worksheet Lw of Cosines.doc QUESTIONS??? If you re finding side, do you hve SS? If Yes, then... If you re finding n ngle, do you hve SSS? If Yes, then... Lw of Cosines (rerrnged) nything else...use your Lw of Sines! 6
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Chp. 3_4 Trigonometry.noteook EXTR PRCTICE TIME...Finish for HW!!! 1) Review Primry Trig Rtios_Lw of Sines_Cosines.pdf Corrections to solutions: #1c) 19.1 #1d) 20.63 scrtch (e) & (f) #4. 877.3 km 2) Puzzle Review Primry Trig, Lw of Sines_Cosines.pdf MORE PPLICTIONS... erings 8
Chp. 3_4 Trigonometry.noteook pplictions: erings NOTE: Ex #1:(p. 122) EX #2: Solving n ppliction question... (p. 166) Compss Rose nimtion HOMEWORK... *** Quiz on Mondy Primry Trig Rtios & Pythgoren Theorem When your finished the quiz... HOMEWORK: More pplictions/word Prolems Pge 154 #5, 6, 9, 10, 11 (erings see exmple from Fridy) Pge 172 #9, 10, 12, 13, 14 Lw of Sines/Cosines Finding ngles/sides/solving/word prolems MORE PRCTICE!!! Puzzle Review Primry Trig, Lw of Sines_Cosines.pdf Solutions to the puzzle... Puzzle Review Solutions.pdf Logic Figure Out The Digits.doc DUE FIRST OF CLSS MONDY Trigonometry Summry ND 'The MIGUOUS Cse'... Wrm Up 1. Determine the mesure of the otuse ngle : miguous Cse 9
Chp. 3_4 Trigonometry.noteook Notes miguous Cse.pdf 4.3 The miguous Cse of the Sine Lw Criteri for the miguous Cse... Must e given SS Given ngle is cute < *** If LL 3 criteri re met, then... CLCULTE THE LTITUDE lt = sin CSE 1: < ltitude there is NO SOLUTION CSE 2: = ltitude there is ONE SOLUTION [Right Tringle] CSE 3: >ltitude this is the 'MIGUOUS CSE'...TWO SOLUTIONS 1) cute Tringle (ngle, θ, is found with Lw of Sines) 2) Otuse Tringle (ngle is 180ο θ) EXPLORE... Two sides in n otuse tringle re 3 m nd 4 m in length. The ngle tht is opposite the 3 m side mesures 40. Determine the mesure of the ngle tht is opposite the 4 m side. SMPLE NSWER MUST MEMORIZE Criteri for the miguous Cse... Must e given SS Given ngle is cute THESE < *** If LL 3 criteri re met, then... NOTES IN ORDER CLCULTE THE LTITUDE lt = sin CSE 1: < ltitude there is NO SOLUTION CSE 2: = ltitude there is ONE SOLUTION [Right Tringle] CSE 3: >ltitude this is the 'MIGUOUS CSE'...TWO SOLUTIONS TO KNOW 1) cute Tringle (ngle, θ, is found with Lw of Sines) 2) Otuse Tringle (ngle is 180ο θ) MIGUOUS ck to the Wrm-Up... CSE 1. Determine the mesure of the otuse ngle : Exmple 2: Exmple 3: Solve the tringle C if = 10, = 12 nd ngle = 72o. C 10
Chp. 3_4 Trigonometry.noteook HOMEWORK... Worksheet miguous Cse.pdf Do questions #1, 2 & 4 MEMORIZE!!! miguous Cse Slide Show.ppt Criteri for the miguous Cse... Must e given SS Given ngle is cute < *** If LL 3 criteri re met, then... CLCULTE THE LTITUDE lt = sin CSE 1: < ltitude there is NO SOLUTION CSE 2: = ltitude there is ONE SOLUTION [Right Tringle] CSE 3: >ltitude this is the 'MIGUOUS CSE'...TWO SOLUTIONS 1) cute Tringle (ngle, θ, is found with Lw of Sines) 2) Otuse Tringle (ngle is 180ο θ) REVIEW: Solving Olique Tringles... The miguous Cse The miguous cse When using the lw of sines to solve tringles, under specil conditions there exists n miguous cse where two seprte tringles cn e constructed (i.e., there re two different possile solutions to the tringle). Summry: miguous Cse When given two sides nd non contined ngle tht is cute the miguous cse must e exmined Given generl tringle C, the following conditions would need to e fulfilled for the cse to e miguous: The ltitude of the tringle will dictte how mny solutions re possile. The only informtion known out the tringle is the ngle nd the sides nd, where the ngle is not the included ngle of the two sides. The ngle is cute (i.e., < 90 ). The side is shorter thn the side (i.e., < ). ( is the ltitude of right tringle with ngle ) The ngle is not right ngle (i.e., > sin ). Given ll of the ove premises re true, the ngle my e cute or otuse mening, one of the following is true: [ ] OR [ ] Exmple 4: MIGUOUS??? 11
Chp. 3_4 Trigonometry.noteook HOMEWORK... Worksheet miguous Cse.pdf Wrm Up Given ΔRST hs ngle R = 58o, r = 48 nd s = 25. Solve the tringle, if there is more thn one possile, solve oth!! #5, 6, & 7 Pge 184: #7, 8, 11 EXMPLE #5: (p. 180) MIGUOUS!!! REVIEW - Trigonometry Pythgoren Theorem & Primry Trig Rtios REMEMER: "SOH CH TO" HYP OPP θ Reference ngle DJ pplictions of Primry Trig Lw of Sines & Its pplictions miguous Cse??? (Lw of Sines - finding n ngle) - given side (), the ngle opposite () nd nother side ()... CSE #1: > -> only one solution CSE #2: = -> only one solution CSE #3: <... Determine the ltitude length (sin) (i) < ltitude -> no solution (ii) = ltitude -> one solution (right tringle) (iii) > ltitude -> two solutions... (miguous Cse) (1) cute ngle (2) otuse ngle (180o - cute) Lw of Cosines & Its pplictions erings nd Multi-step Word Prolems "Solving" - find LL ngles & sides 12
Chp. 3_4 Trigonometry.noteook Review for Test Lots of Prctice from the Textook!!! Chpter Review... Pge 128 (Frequently sked Questions) Pge 153 Pge 174 Pge 199 REVIEW TIME!!! Review Trigonometry.doc Prctice Questions... Pge 129 #1 9 Pge 154 #1 12 Pge 175 #1 9 Pge 200 #1 8 Prctice Tests... Pge 152 #1 8 Pge 198 #1 7 13
ttchments Worksheet Primry Trig Rtios.doc Worksheet pplictions of Primry Trig.doc Worksheet Lw of Cosines.doc Review Primry Trig_Lw of Sines_Cosines.doc Worksheet re of Tringle_Review Trig.doc Review Triggonometry.doc Worksheet Lw of Sines.doc Puzzle Review Primry Trig, Lw of Sines_Cosines.pdf Review Primry Trig Rtios_Lw of Sines_Cosines.pdf Logic Figure Out The Digits.doc Puzzle Review Solutions.pdf Worksheet miguous Cse.pdf Notes miguous Cse.pdf miguous Cse Slide Show.ppt