Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as
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1 Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle, sin nd sin. sin. Sole for. sin nd sin Sustitute. sin sin n e rewritten s Te sme type of rgument n e used to sow tt sin sin sin C sin sin Lw of Sines: Te rtio of te sine of n ngle to te lengt of its opposite side is te sme for ll tree ngles of ny tringle. sin sin sin C or sin sin sinc. C Soling Tringle: finding ll of te missing sides nd ngles Note: Te Lw of Sines n e used to sole tringles gien S nd S. Ex1: Sole te tringle C gien tt 15, C 5, nd 9. 9 C sin sinc We re gien S, so we will use te Lw of Sines. Sole for : Find m using te tringle sum. m sin sin Sole for : C : m, m, m C,,, Pge 1 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
2 miguous Cse: Wen gien SS, tere ould e tringles, 1 tringle, or no tringles tt n e reted wit te gien informtion. Ex: Sole te tringle C (if possile) wen m C 54, 10, 7. Gien SS, use Lw of Sines. sinc sin Sole for. Tere is no possile tringle wit te gien informtion. Ex3: Sole te tringle C (if possile) wen m C 31, 46, 9. Gien SS, use Lw of Sines. Sole for. sinc sin Note tt te lultor only gies te ute ngle mesure for. Tere does exist n otuse ngle wit te sme sine m Tis is lso n pproprite mesure of n ngle in tringle, so tere re tringles tt n e formed wit te gien informtion. Tringle 1 Tringle Pge of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
3 pplition Prolems 1. Drw piture!. Use te Lw of Sines to sole for wt is sked in te prolem. Ex4: Te ngle of eletion to mountin is 3.5. fter driing 13 miles, te ngle of eletion is 9. pproximte te eigt of te mountin. z 3.5 θ 9 13 (not drwn to sle) First, find θ: You Try: Sole te tringle C (if possile) wen m 98, 10, 3. QOD: Explin wy SS is te miguous se wen soling tringles. Pge 3 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
4 Dte: 6. Lw of Cosines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Cosines). Lw of Cosines: For ny tringle, C C osc os os Note: In rigt tringle, Teorem) os90 0 (Pytgoren Te Lw of Cosines n e used to sole tringles wen gien SS or SSS. Ex1: Sole te tringle C wen m 49, 4, & 15. Note: Te gien informtion is SS. Use os. Now tt we e mting pir of side nd ngle, we n use te Lw of Sines. or sin sin Now find te two possiilities for m C using te tringle sum: Sine is te sortest side, it must e opposite te smllest ngle. So m C. Ex: Sole te tringle C wen 31, 5, & 8. Note: Te gien informtion is SSS. Use os. (Or ny of tem!) Now tt we e mting pir of side nd ngle, we n use te Lw of Sines. or sin sin Pge 4 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
5 pplition Prolems 1. Drw piture!. Use te Lw of Cosines to sole for wt is sked in te prolem. Ex3: plne tkes off nd trels 60 miles, ten turns 15 nd trels for 80 miles. How fr is te plne from te irport? (not drwn to sle) Using te piture, we n find te ngle in te tringle to gie us SS. Use te Lw of Cosines: osc re of Tringle Rell: 1 sin sin ; 1 sin Formul for te re of Tringle Gien SS: θ 1 sinc Ex4: Find te re of tringle C sown C We will use 1 sin. Heron s re Formul Semi-Perimeter: s re of Tringle Gien SSS: s s s s Pge 5 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
6 Ex5: Find te re of te tringle wit side lengts 5 m, 6 m, nd 9 m Semiperimeter: s s s 10 s s s s You Try: Two sips lee port wit 19 ngle etween teir plnned routes. If tey re treling t 3 mp nd 31 mp, ow fr prt re tey in 3 ours? QOD: Cn tere e n miguous se wen using te Lw of Cosines? Explin wy or wy not. Pge 6 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
7 Dte: 6.3 Vetors in te Plne Syllus Ojeties: 5.1 Te student will explore metods of etor ddition nd sutrtion. 5. Te student will deelop strtegies for omputing etor s diretion ngle nd mgnitude gien its oordintes. 5.4 Te student will resole etors into unit etors. 5.7 Te student will sole rel-world pplition prolems using etors in two nd tree dimensions. Direted Line Segment: segment wit diretion nd distne : Initil Point (strt); : Terminl Point (end) Coordintes of : x1, y 1 Coordintes of : x, y Vetor (): Vetors e Distne nd Mgnitude! Note: Equilent mens sme mgnitude nd diretion. Mgnitude (lengt) of Direted Line Segment : Note: Tis is te distne formul! x x1 y y 1 Component Form of Vetor: ( x x1, y y 1 lwys strt t te terminl side nd sutrt te initil side so te omponent etor will e going te rigt diretion. Ex1: Grp te etor 3, nd find te mgnitude. Note: 3, ould e pled nywere on te oordinte grid. Ple it in stndrd position, wi is wit te initil point t te origin. Mgnitude: Note: If etor u is written in omponent form, u,, ten te mgnitude of u is u. Tis is euse te initil point is te origin, 0,0. Vetor ddition: Let u u1, u nd 1,. Ten u u1 1, u. Ex. 5, 3, 8 Slr Multiplition: Let u u1, u nd k e ny onstnt. Ten ku ku1, ku. Ex. 3 3,5 Note: If k 0, ten ku is in te opposite diretion. Pge 7 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
8 Ex4: Use te grp of te etors to omplete e exmple elow. w u ) Sow tt u. Sow tt u. u Sow tt te diretion of u is te sme s te diretion of. Use slope: Diretion of u = ; diretion of = Te diretion nd mgnitude re te sme, so u. ) Find te omponent form nd te mgnitude of und w. Component form of u: u u (see oe) Component form of w: w w u 3w ) Find te omponent form of u 3w. Unit Vetor: etor wit mgnitude of 1 unit etor in te diretion of etor n e found y diiding y te mgnitude of. Unit Vetor in te Diretion of : Stndrd Unit Vetors: unit etors i nd j in stndrd position long te positie x- nd y-xes i 1,0 & j 0,1 ny etor n e written in terms of te stndrd unit etors. Ex5: Write te etor,5 in terms of te stndrd unit etors. Pge 8 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
9 Ex6: Find unit etor in te diretion of te gien etor. Verify your nswer is unit etor nd gie your nswer in omponent form nd stndrd unit etor form. i 4j Find te mgnitude: i 4j Diide te originl etor y its mgnitude: Verify mgnitude of unit etor: Rell: In te unit irle, x os, y sin. Tis leds into noter wy of expressing etor, in terms of its diretion ngle, θ. Diretion ngle: in stndrd position, te ngle te etor mkes wit te positie x-xis (ounterlokwise) Resoling Vetor: in terms of its diretion ngle, θ, etor n e written s u os, u sin Mgnitude: Ex8: Find te mgnitude nd diretion ngle of i 6j. Diretion ngle: Pge 9 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
10 Ex9: Find te omponent form of gien its mgnitude nd its diretion ngle. 5, 30 os i sin j pplition: Resultnt Fore Ex10: Two fores t on n ojet: u 3, u 45 nd 4, 30 mgnitude of te resultnt fore.. Find te diretion nd Write e etor in omponent form: Te resultnt fore is te sum u : pplition: ering Ex11: plne flies due est t 500 km/ nd tere is 60 km/ wit ering of 45. Find te ground speed nd te tul ering of te plne. Sket digrm: Refletion: (You Try) Find te omponent form of gien its mgnitude nd te ngle it mkes wit te positie x-xis., diretion: i 3j Pge 10 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
11 Dte: 6.4 Vetors nd Dot Produt Syllus Ojetie: 5.3 Te student will explore metods of etor multiplition. 5.5 Te student will determine if two etors re prllel or perpendiulr (ortogonl). 5.6 Te student will derie n eqution of line or plne y using etor opertions. 5.7 Te student will sole rel-world pplition prolems using etors in two nd tree dimensions. Dot Produt: Let u u1, u nd 1,. Te dot produt is u u1 1 u. Note: Te dot produt is slr. Ex1: Elute 5, 3,4. Properties of te Dot Produt: 1. u u. u u = u 3. 0 u 0 4. u + w = u u w 5. u u u Ex: Elute te following gien u 3, 6 ; 1, 0 ; w 5, () ww () w () w u (d) u w u ngle etween Two Vetors: os u u os 1 u u u θ u Lw of Cosines: Property of Dot Produt: Proof: Use te tringle. u u u os u u u u os Expnd: Property of Dot Produt: u u u u u u os u u u u os Property of Equlity: u u os os u u Pge 11 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
12 os Ex3: Find u, were θ is te ngle etween u nd. u u u 6, 8, 5 6 pplition Ex4: Find ngle of te tringle wit erties 3,0, 4,, 5,1. C Ortogonl Vetors: two etors wose dot produt is equl to 0 Or ngle etween two non-zero ortogonl etors is 90 Prllel Vetors: two etors wose os = 1 or 1, or ngle etween is 0,180, or 360 u u must = 1 or -1 Ex5:.) re te etors ortogonl, prllel, or neiter? 3i j, w 3i 4j Find w: Te etors re..) 3, nd w 4,6 Te etors re. Pge 1 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
13 u Vetor Projetion: te projetion of u onto is denoted y: proj u u u = u u 1 u 1 = proj u Ex6: Find te projetion of onto w. Ten write s te sum of two ortogonl etors, wit one te proj w. 1,3 ; w 1,1 proj w w w w proj w Pge 13 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
14 pplition: Fore Ex7: Find te fore required to keep 00-l rt from rolling down 30 inline. Drw digrm nd lel: Te fore due to grity: g 00j (grity ts ertilly downwrd) Inline etor: 3 1 os30 i sin 30 j i j g Fore etor required to keep te rt from rolling: f proj g f g Mgnitude of Fore: f 00 f 30 pplition: Work W os fore distne Ex8: person pulls wgon wit onstnt fore of 15 ls t onstnt ngle of 40 for 500 ft. Wt is te person s work? 40 w os ls 500 ft You Try: Find te projetion of onto u. Ten write s te sum of two ortogonl etors, wit one te proj u. i 3 j; u i j QOD: If u is unit etor, wt is uu? Explin wy. Pge 14 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
15 Syllus Ojeties: 7.1 Te student will grp omplex numer on te omplex/rgnd plne. 7. Te student will represent omplex numer in trigonometri (polr) form. 7.3 Te student will simplify expressions inoling omplex numers in trigonometri (polr) form. 7.4 Te student will ompute te powers of omplex numers using DeMoire s Teorem nd find te nt roots of omplex numer. Complex Numer Plne (rgnd Plne): orizontl xis rel xis; ertil xis imginry xis +i Plotting Points in te Complex Plne Ex1: Plot te points 3 4 i, 1 3 i, & C i in te omplex plne. C solute Vlue (Modulus) of Complex Numer: te distne omplex numer is from te origin on te omplex plne i (Tis n e sown using te Pytgoren Teorem.) Ex: Elute 3 i. Rell: Trigonometri form of etor: u os,sin Trigonometri Form of Complex Numer z = + i: z r os i sin Note: Tis n lso e written s z r is. r os, rsin, r tn r = modulus; θ = rgument Writing Complex Numer in Trig Form Ex3: Find te trigonometri form of 3i. 1. Find r: r. Find θ: tn tn 3. Find Qudrnt: Creful wit finding te ngle! 4. z r(os i sin ) Pge 15 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
16 Writing Complex Numer in Stndrd Form ( + i) Ex4: Write 9is in stndrd form. Expnd: 9is 9 os i sin Multiplying nd Diiding Complex Numers Let z 1 r 1 os 1 i sin 1 nd z r os i sin. Multiplition: z1 z r1 r os 1 i sin 1 z1 r1 Diision: z r os i sin 1 1 z1 4 os i sin, z os i sin Ex5:. Express te produt of z 1 nd z in stndrd form.. find te rtio of z 1 nd z n n Powers of Complex Numer: De Moire s Teorem z r os i sin r os n i sin n n Ex5: Elute 5 i. Rewrite in trig form: n t Roots of Complex Numer: n n k k n k z r os i sin ris n n n n n, k 0,1,,... n 1 Note: Eery omplex numer s totl of n n t roots. Ex: Find te ue roots of 8i. Write in trig form: Elute te roots: Pge 16 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
17 Roots of Unity: te n t roots of 1 Ex6: Express te fift roots of unity in stndrd form nd grp tem in te omplex plne. 5 t Roots of Unity: 5 1 0i r 1, 0 1is k k 1is0 1is is 5 5 You Try: 1. Write e omplex numer in trigonometri form. Ten find te produt nd te quotient. 1 3 i, 3i. Sole te eqution x (You sould e 4 solutions!) QOD: Is te trigonometri form of omplex numer unique? Explin. Pge 17 of 17 Prelulus Grpil, Numeril, lgeri: Lrson Cpter 6
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