4.3 The Sine Law and the Cosine Law
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1 4.3 Te Sine Lw nd te osine Lw Te ee Tower is te tllest prt of nd s rliment uildings. ronze mst, wi flies te ndin flg, stnds on top of te ee Tower. From point 25 m from te foot of te tower, te ngle of elevtion of te top of te tower is From te sme point, te ngle of elevtion of te top of te mst is INVESTIGTE & INQUIRE To find te eigt of te mst, use te digrm sown. Te given informtion is mrked on te digrm. is te eigt of te mst, nd is te eigt of te tower Find te eigt of te mst, to te nerest tent of metre, using only rigt tringles m 2. ) List mesurements you would use to find te eigt of te mst using te osine lw in. ) Find tese mesurements. ) Use te osine lw to find te eigt of te mst, to te nerest tent of metre. 3. ) List mesurements you would use to find te eigt of te mst using te sine lw in. ) Find tese mesurements. ) Use te sine lw to find te eigt of te mst, to te nerest tent of metre. 4. ompre your nswers from questions 1, 2, nd 3. Wi metod did you prefer? Eplin. 4.3 Te Sine Lw nd te osine Lw MHR 283
2 You ve previously pplied te sine lw nd te osine lw to ute tringles. You ve seen tt te sine lw nd te osine lw lso pply to otuse tringles. Te sine lw for ute nd otuse tringles n e developed s follows. In, drw perpendiulr to, or to etended. is te ltitude or eigt,, of. ute Tringle Otuse Tringle (180 ) In, = sin In, = sin(180 ) Rell tt sin (180 θ) = sin θ. = sin = sin = sin In, = sin In, = sin = sin For ot te ute nd te otuse tringles, sin = sin sin sin ivide ot sides y : = = sin sin sin Simplify: = y drwing te ltitude from, we n similrly sow tt sin sin = omining te results gives te following forms of te sine lw. sin sin sin = = = = sin sin sin 284 MHR pter 4
3 EXMLE 1 Te Sine Lw, Given Two ngles nd Side In RST, S = 40, T = 21, nd r = 46 m. Find t, to te nerest entimetre. SOLUTION rw digrm. Find te mesure of R. R = = 119 Use te sine lw to find t. t r = sin T sin R S 40 R 46 m 21 T t sin 21 = 46 sin sin 21 t = sin 119 t = 19 So, t = 19 m, to te nerest entimetre. EXMLE 2 Te Sine Lw, Given Two Sides nd te ngle Opposite One of Tem In QR, = 105.2, p = 23.2 m, nd r = 18.5 m. Solve te tringle, rounding te side lengt to te nerest tent of entimetre nd te ngles to te nerest tent of degree, if neessry. SOLUTION rw digrm. Use te sine lw to find te mesure of R m sin R r = sin p Q 23.2 m R sin R sin = sin sin R = 23.2 R = Te Sine Lw nd te osine Lw MHR 285
4 Find te mesure of Q. Q = = 24.5 Use te sine lw to find q. q p = sin Q sin q sin = sin sin 24.5 q = sin = 10.0 In QR, R = 50.3, Q = 24.5, nd q = 10.0 m. Te osine lw for ute nd otuse tringles n e developed s follows. In, drw perpendiulr to, or to etended. is te ltitude or eigt,, of. ute Tringle In, = os = os nd 2 = In, 2 = 2 + ( ) 2 = = 2 + ( ) 2 2 = os 286 MHR pter 4
5 Otuse Tringle (180 ) + In, = os (180 ) = os (180 ) Rell tt os (180 θ) = os θ. = os nd 2 = In, 2 = 2 + ( + ) 2 = = 2 + ( ) + 2 = ( os ) 2 = os Te forms of te osine lw re s follows. 2 = os os = 2 2 = os os = 2 2 = os os = 2 EXMLE 3 Te osine Lw, Given Tree Sides In, = 9.6 m, = 20.6 m, nd = 14.7 m. Solve te tringle. Round e ngle mesure to te nerest tent of degree. 4.3 Te Sine Lw nd te osine Lw MHR 287
6 SOLUTION rw digrm. Use te osine lw to find te mesure of n ngle os = = 2(9.6)(14.7) = m 20.6 m 14.7 m Use te sine lw to find te mesure of. sin sin = sin sin = sin sin = 20.6 = 25.1 Find. = = 40.6 In, = 25.1, = 114.3, nd = EXMLE 4 Te osine Lw, Given Two Sides nd te ontined ngle Find te lengt of, to te nerest tent of metre. E m F 3.9 m 288 MHR pter 4
7 SOLUTION Use EF nd te osine lw to find te lengt of E. E 2 = F 2 + EF 2 2(F)(EF)os F = (3.9)(2.5)os 97.4 E = 4.9 Use E to find te mesure of. = = 75.1 Use E nd te sine lw to find te lengt of. 4.9 = sin 56.7 sin sin 56.7 = sin 75.1 = 4.2 = 4.2 m, to te nerest tent of metre. Key onepts Te forms of te sine lw re = = sin sin sin sin sin sin = = Te sine lw n e used to solve ny tringle wen given ) te mesures of two ngles nd ny side ) te mesures of two sides nd te ngle opposite one of tese sides Te forms of te osine lw re 2 = os os = 2 2 = os os = 2 2 = os os = 2 Te osine lw n e used to solve ny tringle wen given ) te mesures of two sides nd te ontined ngle ) te mesures of tree sides 4.3 Te Sine Lw nd te osine Lw MHR 289
8 ommunite Your Understnding 1. esrie ow you would solve e of te following tringles. Justify your osen metod. ) ) 24 7 m 8 m 18 m E m ) U d) X 10 m m 33 m Y 114 T 18 m S F Z 2. Eplin wy you nnot strt wit te sine lw to solve XYZ. 31 m X m Y Z 3. Eplin wy you nnot strt wit te osine lw to solve KLM. K 3.5 m L M rtise 1. Find te lengt of te indited side, to te nerest tent. ) R ) E 31.2 s 15.2 g 82 G 63 T S F ) d) E F f 290 MHR pter 4
9 e) f) M K k 41.1 L Q R 101 r 8.3 d) In UVW, W = 123.9, V = 22.2, v = 27.5 km. e) In XYZ, X = 92.3, y = 3.1 m, z = 2.8 m. f) In FGH, f = 12.6 m, g = 8.5 m, = 6.3 m. 2. Find te mesure of te indited ngle, to te nerest tent of degree. ) ) 27.3 K J L Find te lengt of te indited side, to te nerest tent. ) 13 m ) X d) L ) 16 mm 17.5 Y Z N 16.8 M 21 mm mm e) R f) Q Solve e tringle. Round nswers to te nerest tent, if neessry. ) In, = 84, = 40, = 5.6 m. ) In QR, R = 28.5, p = 10.4 m, r = 6.3 m. ) In LMN, M = 62, l = 16.9 m, n = 15.1 m. G F E ) d) 10 m m Q 51.8 R S 4.3 Te Sine Lw nd te osine Lw MHR 291
10 e) E 9.5 m F f) m 50.1 G H 5. Find te mesure of te indited ngle, to te nerest tent of degree. ) ) 4.3 m 2.7 m θ m 12.3 m θ m 9.8 m 7.3 m Q 6.5 m 84.3 S 91.7 ) d) 25 θ 10 m 20 m 100 m 104 θ m m R pply, Solve, ommunite 6. Solve. Round nswers to te nerest tent. 7. Mesurement n isoseles tringle s two 5.5-m sides nd two 32.4 ngles. Find ) te perimeter of te tringle, to te nerest tent of entimetre ) te re of te tringle, to te nerest tent of squre entimetre 8. Inquiry/rolem Solving irport X is 150 km est of irport Y. n irrft is 240 km from irport Y, nd 23 nort of due west from irport Y. How fr is te irrft from irport X, to te nerest kilometre? 292 MHR pter m
11 9. pplition To determine te eigt of te ee Tower on rliment Hill in Ottw, mesurements were tken from seline. It ws found tt = 50 m, XY = 42.6, X = 60, nd X = lulte te eigt of te ee Tower, to te nerest metre. Y We onnetion To lern more out te istory nd onstrution of te rliment uildings, visit te ove we site. Go to Mt Resoures, ten to MTHEMTIS 11, to find out were to go net. Write rief report. X 10. Sip nvigtion Two sips left ort Hope on Lke Ontrio t te sme time. One trvelled t 12 km/ on ourse of 235. Te oter trvelled t 15 km/ on ourse of 105. How fr prt were te sips fter four ours, to te nerest kilometre? 270 N Mesurement Find te re of XYZ, to te nerest tent of squre metre m 180 X 6.8 m 12. ommunition ) Use te osine lw to find, to te nerest tent. ) Use te ytgoren teorem to find, to te nerest tent. ) Eplin wy te two metods give te sme results in rigt tringle. Z m 5.2 Y 13. Sine lw in rigt tringles Rigt is sown. Write e of te rtios,, nd in terms of,, or, nd sin sin sin verify tt = = for rigt tringle. sin sin sin 4.3 Te Sine Lw nd te osine Lw MHR 293
12 14. Stikine nyon Te Stikine nyon in entrl ritis olumi is often referred to s nd s Grnd nyon. Two points X nd Y re sigted from seline of lengt 30 m on te opposite side of te nyon. Te ngle mesurements reorded from positions nd were XY = 31.3, XY = 18.5, X = 25.6, nd Y = Find te distne from X to Y, to te nerest metre. X Y 15. Geometry Use te osine lw to sow tt opposite ngles in prllelogrm re ongruent. 16. Mesurement In RST, RS = 4.9 m, ST = 3.7 m, nd RT = 8.1 m. Find te re of RST, to te nerest tent of squre metre. 17. Mesurement In, = 46 m, = 42.2, nd = Find te re of, to te nerest tent of squre metre. 18. Mesurement Find te volume of te rigt prism, to te nerest ui entimetre. 19. Mesurement Find te volume of te rigt prism, to te nerest ui metre. 8.8 m m 15 m m 9.4 m 16.2 m 20. nlyti geometry QR s verties (1, 5), Q(6, 7), nd R( 2, 1). Find te ngle mesures, to te nerest tent of degree. HIEVEMENT ek Knowledge/Understnding Tinking/Inquiry/rolem Solving ommunition pplition n equilterl tringle s een resed nd folded so tt its verte now rests on t, su tt = 1 nd = 2. Find te lengt of ) ) Q ) Q Q MHR pter 4
13 REER ONNETION Surveying Surveying is te sientifi mesurement of nturl or rtifiil fetures on te Ert s surfe. Surveyors re involved in wide vriety of tsks tt require very urte knowledge of lotions. Te distnes nd ngles determined y surveyors re used in mny wys, inluding drwing mps, positioning uildings nd oter strutures orretly, nd defining te property lines tt seprte one piee of lnd from noter. euse nd is te world s seond-lrgest ountry, surveying nd s een n enormous tsk. For emple, it took lmost 60 yers to omplete survey of te nd-u.s order, prt of wi runs troug four of te Gret Lkes. s result of over 150 yers of surveying work, detiled mps now eist for ll prts of nd. 1. From point, te distne to one end of pond is 450 m nd te distne to te oter end is 520 m. Te ngle formed y te lines of sigt is 115. Find te lengt of te pond, to te nerest ten metres. 450 m m 2. Reser Use your reser skills to investigte te following. ) te edution nd trining required to eome surveyor, nd te orgniztions tt employ surveyors ) te use of different types of surveying equipment, inluding mnully ontrolled, eletroni, nd potogrpi instruments, nd te use of stellite tenology ) te work of te Geologil Survey of nd in eploring nd mpping te ountry TTERN ower ) opy nd omplete te pttern = = = ) esrie te pttern in words. ) Eplin wy te pttern works. d) Write te net 2 lines of te pttern. e) Use te pttern to find Te Sine Lw nd te osine Lw MHR 295
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