UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4 Solving for ngles Pg. 99 #, 6ce, 8c, 9, 10, 1 My 1 5.4 (51) 5. Specil ngles Pg. 87 # 4, 7, 9, 11 My 5.5 (5) 5.6 5.7 Solving for Otuse ngles QUIZ (5.1 5.3) Pg. 318 # 1, 4 Pg. 36 # 3, 4 My 3 5.6 (53) The miguous se WS 5.6 Pg. 318 # 3, 5 My 4 5.7 (54) 5.6 5.7 Trig Word Prolems WS 5.7 # 1-10 My 7 5.8 (55) 5.6 5.7 More Trig Word Prolems QUIZ (5.4-5.6) WS 5.7 # 1 16 Pg. 319 # 7, 8 Pg. 36 # 5, 6, 7, 10 My 8 5.9 (56) 5.8 3D Trig Word Prolems/Inccessile Distnces WS 5.9 Q # 5 DIRETLY OVE Pg. 33 # 3, 4, 5, 9 My 9 5.10 (59) Review for Unit 5 Test Pg. 338 # 1 5, 8 1, Pg. 340 # 1,, 3, 6, 7, 8 My 11 5.11 (60) UNIT 5 TEST
MR3U Lesson 5.1 Trigonometry Review 4 5 sin cos sin cos 3 3 tn tn 1. Determine length x in ech tringle. Round your nswer to one deciml plce. ) 8.4 cm 15 x ) 8. km x 4. Determine the mesure of, to the nerest degree. 1.1 cm 8. cm, 8. cm, 75, c 10. 1cm. Determine the mesure of 3. In.
4. Solve the following tringles. Q 15 cm p 41 P 3.8 m 5.4 m 6.0 cm 6 cm R 60 6.5 m 65 8 cm 8.0 cm HW: WS 5.1
Trig Formul Sheet Right tringle opposite hypotenuse djcent sin cos tn opposite hypotenuse djcent hypotenuse opposite djcent csc sec cot hypotenuse opposite hypotenuse djcent djcent opposite c c cute nd Otuse Tringles: The Sine Lw: c In ny, or sin sin c (when solving for sides) sin sin sin sin (when solving for ngles) c The osine Lw Use this version when given sides nd contined ngle. c c c c ccos ccos cos When 3 sides re known the osine Lw cn e written s follows: cos c c cos c c cos c
MR3U Lesson 5. Relted ngles Drw ech of the following ngles in stndrd position, clerly showing the R (Relted cute ngle). The R is the ngle tht the terminl rm mkes with the x xis. You must drw the rottion direction of the ngle nd lel it for it to e considered the desired ngle. 130 10 50 310 140 30 340 80 For ese of clcultion, ngles cn e plced on the rtesin plne within unit circle (circle with rdius of 1 unit). Sketch 50 ngle in the unit circle elow y (0,1) Note tht in circle with rdius r, In the unit circle (r = 1), sin sin (-1,0) (1,0) x cos cos tn tn (0,-1)
Now we will move this tringle to ech of the four qudrnts: Sketch ech of the following ngles in stndrd position. lerly show the R (relted cute ngle) nd lel the point on the terminl rm which intersects the unit circle. 130 30 310 y (0,1) y (0,1) y (0,1) (-1,0) (1,0) (-1,0) (1,0) (-1,0) (1,0) x x x (0,-1) (0,-1) (0,-1) 50 130 30 310 sin cos tn ST Rule: We sy tht (R). 50, 130, 30, nd 310 re relted ngles, nd tht 50 is the relted cute ngle List the relted ngles for: ) 0 ) 0
Given the vlue of one trig rtio, provide 3 relted evluted trig rtios. ) sin10 0. 1736 ) cos47 0. 680 ) tn 70. 7475 d) sin0 0. 648 For ech of the following trig expressions write n equivlent trig expression. Use ngles etween 0 nd 360. ) sin 50 ) cos 110 c) tn 345 d) cos 80 e) sin( 160) f) tn( 95) g) cos 80 = h) tn 70 = i) sin 305 = j) sin 10 = k) tn 170 = l) cos 15 = m) cos 0 = n) tn 306 = o) sin 13 = p) tn( 100) = q) cos( 15) = r) sin( 305) = OR = OR = OR =
MR3U Lesson 5.3 Solving For ngles Solve ech of the following to the nerest degree, where 0 360. ) sin 0. 6 ) cos 0. c) tn 1. 966 d) sin 0. 7547
Outside The Unit ircle 1., 4 3 lies on the terminl rm of ngle in stndrd position. ) Drw sketch of ngle. ) Determine the primry trig rtios for ngle. c) lculte the vlue of to the nerest degree. y 5 4 3 1 5 4 3 1 1 3 4 5 x 1 3 4 5. is third qudrnt ngle such tht sin 0. 7 ) Drw sketch of ngle. ) Determine the primry trig rtios for ngle. c) lculte the vlue of (to the nerest degree) using oth counterclockwise nd clockwise rottion. y x HW: p. 99 #, 6ce, 8c, 9, 10, 1
MR3U Lesson 5.4 Specil ngles The Unit ircle (cos, sin ) Ex. Evlute ech of the following: (exct nswers required) ) tn 45 sin 45 ) tn 30 cos 30 c) sin 60 cos 45
Ex. Evlute ech of the following. nswers must e exct. No clcultors permitted. sin 10 cos 180 tn 30 sin 315 cos 10 tn 300 sin 70 cos 60 sin 40 tn 5 tn 0 cos 135 sin 150 sin 300 cos 330 tn 150 Ex. Solve ech of the following, where 0 360. ) ) c) 3 sin or cos or.. 1 sin or. d) tn 3 or. e) cos 0 or. f) 3 cos or. g) tn 1 or. h) sin 1 or. TEXT HW: Pg. 87 #4,7,9,11
MR3U Lesson 5.5 Solving Otuse Tringles 1. Find the vlue of, correct to the nerest degree..5 cm 5.5 cm 3.5 cm. Solve the following tringle: (round ngle mesures to 1 deciml plce) For ech of the following, drw digrms (somewht to scle would e nice) 3. Solve 4. Solve 5. Solve, where 48, 5. 0cm, nd c 6. 3, where 30,. 0 cm, nd 5. 0, where 30, 7. 0 cm, nd 5. 0 cm. cm. cm. 6. Solve, where 30, 4. 0 cm, nd 5. 0 cm. Pg. 318 # 1, 4 Pg. 36 # 3, 4
MR 3U Lesson 5.6 The miguous se ny tringle without right ngle is clled n olique tringle. The cosine lw nd the sine lw cn e used to determine ngles or sides in ll tringles (cute, right or otuse). In, with sides,, nd c, The Sine Lw The osine Lw c sin sin sin c c cos sin sin sin c c cos c c To solve n olique tringle, you need to know the mesure of t lest one side nd two other prts of the tringle. There re four cses in which this cn hppen. Given Informtion Wht cn e found Lw Required 1. Two ngles nd ny side (S or S) side Sine lw. Two sides nd the contined ngle (SS) side osine lw 3. Three sides (SSS) ngle osine lw 4. Two sides nd the ngle opposite one of them (SS) ngle Sine lw se 4 is clled the miguous cse ecuse sometimes it is possile to drw more thn one tringle for the given informtion. In this cse, there re four possile outcomes if is cute nd two possile outcomes if is otuse. These possiilities re shown for the given nd the given sides nd, in. The side opposite the given ngle is lwys nd sin represents the possile height of the tringle. Determining whether given ngle () is cute or otuse, nd then compring the size of,, nd sin llows you to see which sitution you re deling with nd, in turn, the numer of possile solutions.
Is it relly tringle? tringle with ngle of 63.9, side = 8.3 cm, nd side = 6.9 cm cnnot exist! So how cn you tell? If we were to drw the "tringle" we could immeditely tell tht the sides don't meet! However there is more convenient wy. height of = sin If < sin then there is no possile tringle. Lets check tht reltionship here. h = sin sin(63.9) x 8.3 = 7.4 > 6.8 Since the clculted vlue is greter thn the length of side, there is no possile tringle. Now we hve n esy wy to check. if SS or SS miguous cse
90 (cute) onditions Numer nd Type of Tringles Possile sin sin NoTringle NoTringle ie: side is shorter thn the ltitude of the tringle. sin sin One Right Tringle sin sin one cute Two Tringles one otuse 1 sin one tringle 90 (otuse) onditions Numer nd Type of Tringles Possile NONE > ONE OTUSE TRINGLE
Ex. 1 For, = 3.0 cm, c = 5.0 cm nd = 30, solve the tringle(s). Two sides nd n ngle opposite one side re given (SS) This is the miguous cse of the sine lw, where the given ngle is cute. check the reltion ship etween, c, nd csin. Ex. DEF is given with D = 130, d = 50.0 cm nd e = 0.0 cm. Solve DEF. Two sides nd n ngle opposite one side re given (SS) This is the miguous cse of the sine lw, where the given ngle is otuse. WS 5.6 nd Pg. 318 #3,5
MR 3U Lesson 5.7 Solving Prolems using Trigonometry 1. From window, ldder extends down to the ground with n ngle of depression of 65. The se of the ldder is 4.8 m from the uilding. ) How high is the window? ) How long is the ldder?. From n oservtion tower the ngle of elevtion of wether lloon is 68. In the sme plne, on the other side of the lloon 35.0 km wy, the lloon is sighted from nother loction with n ngle of elevtion of 47. lculte the distnce from the wether lloon to the oservtion tower. WS 5.7 # 1-10
WS 5.7 # 1 16 Pg. 319 # 7, 8 Pg. 36 # 5, 6, 7, 10
MR3U Lesson 5.9 Inccessile Distnces / 3 Dimensions Ex. 1. To find the height of cliff tht is inccessile, surveyor mesures seline of 400 m. In the horizontl plne, 7 nd 35. In the verticl plce T, T 18. Determine the cliff height to the nerest metre. (67 m). From the top of 10 m fire tower, fire rnger oserves fires on the ground elow (not in the sme line of sight). One fire hs n ngle of depression of 6 nd the other hs n ngle of depression of 3. The ngle etween the lines of sights is105. lculte the distnce etween the two fires (to the nerest metre). (811 m) 3. To determine the height (from se X to top Y) of the Pece Tower in Ottw, mesurements were tken from seline of length 50 metres. It ws found tht XY 4. 6, X 60, nd X 81. 65. lculte the height of the Pece Tower to the nerest metre. (73 m) 4. The crows-nest of ycht is 50.0 m ove the wter level. The ngle of depression from the crows-nest to uoy due west of the ot is W of the ycht is 34. How fr prt re the uoys? (7.3 m) 40. The ngle of depression to nother uoy S 70 5. Two rods intersect t 34. Two crs leve the intersection on different rods t speeds of 80 km/h nd 100 km/h. fter h, trffic helicopter which is ove nd etween the two crs tkes redings on them. The ngle of depression to the slower cr is 0 nd the distnce to it is 100 km. How fr is the helicopter from the fster cr? (38.7 km) 6. Jennifer nd lex were flying hot ir lloon when they decided to clculte the stright line distnce from everton to Tndy. From height of 340 m they recorded the ngles of depression of everton nd Tndy s nd 3 respectively. The ngle etween the line of sights to the two towns ws 80. Find the distnce from everton to Tndy. (10.7 km) Pg. 33 # 3, 4, 5, 9