Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as B

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1 Date: 6.1 Law of Sines Syllaus Ojetie: 3.5 Te student will sole appliation prolems inoling triangles (Law of Sines). Deriing te Law of Sines: Consider te two triangles. a C In te aute triangle, sin and sin. a In te otuse triangle, sin sin. a Sole for. sin and asin sin sin Sustitute. asin sin an e rewritten as a sin sin sin Te same type of argument an e used to sow tat C. a Law of Sines: Te ratio of te sine of an angle to te lengt of its opposite side is te same for all tree angles of any triangle. sin sin sin C or a a sin sin sinc C a C a Soling a Triangle: finding all of te missing sides and angles Note: Te Law of Sines an e used to sole triangles gien S and S. Ex1: Sole te triangle C 9 a C gien tat 15, C 5, and 9. We are gien S, so we will use te Law of Sines. sin sinc Sole for : Find m using te triangle sum. m sin sin Sole for a: a C : m, m, mc, a,, Page 1 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

2 miguous Case: Wen gien SS, tere ould e triangles, 1 triangle, or no triangles tat an e reated wit te gien information. Ex: Sole te triangle Gien SS, use Law of Sines. C (if possile) wen mc 54, a 10, 7. sinc sin a Sole for. Tere is no possile triangle wit te gien information. Ex3: Sole te triangle Gien SS, use Law of Sines. Sole for. C (if possile) wen mc 31, 46, 9. sinc sin Note tat te alulator only gies te aute angle measure for. Tere does exist an otuse angle wit te same sine. m Tis is also an appropriate measure of an angle in a triangle, so tere are triangles tat an e formed wit te gien information. Triangle 1 Triangle Page of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

3 ppliation Prolems 1. Draw a piture!. Use te Law of Sines to sole for wat is asked in te prolem. Ex4: Te angle of eleation to a mountain is 3.5. fter driing 13 miles, te angle of eleation is 9. pproximate te eigt of te mountain. z 3.5 θ 9 13 (not drawn to sale) First, find θ: You Try: Sole te triangle C (if possile) wen m 98, 10, 3. QOD: Explain wy SS is te amiguous ase wen soling triangles. Page 3 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

4 Date: 6. Law of Cosines Syllaus Ojetie: 3.5 Te student will sole appliation prolems inoling triangles (Law of Cosines). Law of Cosines: For any triangle, C C a aosc a a os a os a Note: In a rigt triangle, Teorem) a aos90 a a 0 a (Pytagorean Te Law of Cosines an e used to sole triangles wen gien SS or SSS. Ex1: Sole te triangle C wen m 49, 4, & 15. Note: Te gien information is SS. Use a os. Now tat we ae a mating pair of a side and angle, we an use te Law of Sines. a or sin sin Now find te two possiilities for m C using te triangle sum: Sine is te sortest side, it must e opposite te smallest angle. So mc. Ex: Sole te triangle C wen a 31, 5, & 8. Note: Te gien information is SSS. Use a os. (Or any of tem!) Now tat we ae a mating pair of a side and angle, we an use te Law of Sines. a or sin sin Now find te two possiilities for m C using te triangle sum: Sine is te sortest side, it must e opposite te smallest angle. So mc. Page 4 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

5 ppliation Prolems 1. Draw a piture!. Use te Law of Cosines to sole for wat is asked in te prolem. Ex3: plane takes off and traels 60 miles, ten turns 15 and traels for 80 miles. How far is te plane from te airport? (not drawn to sale) Using te piture, we an find te angle in te triangle to gie us SS. Use te Law of Cosines: a a osc rea of a Triangle 1 Reall: sin sin ; 1 sin Formula for te rea of a Triangle Gien SS: θ 1 asinc Ex4: Find te area of triangle C sown C We will use 1 a sin. Heron s rea Formula Semi-Perimeter: a s rea of a Triangle Gien SSS: ss as s Page 5 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

6 Ex5: Find te area of te triangle wit side lengts 5 m, 6 m, and 9 m. a Semiperimeter: s s s 10 ss as s You Try: Two sips leae port wit a 19 angle etween teir planned routes. If tey are traeling at 3 mp and 31 mp, ow far apart are tey in 3 ours? QOD: Can tere e an amiguous ase wen using te Law of Cosines? Explain wy or wy not. Page 6 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

7 Date: 6.3 Vetors in te Plane Syllaus Ojeties: 5.1 Te student will explore metods of etor addition and sutration. 5. Te student will deelop strategies for omputing a etor s diretion angle and magnitude gien its oordinates. 5.4 Te student will resole etors into unit etors. 5.7 Te student will sole real-world appliation prolems using etors in two and tree dimensions. Direted Line Segment: a segment wit diretion and distane : Initial Point (start); : Terminal Point (end) Coordinates of : x, y Coordinates of : x, y Magnitude (lengt) of a Direted Line Segment : Note: Tis is te distane formula! x x y y Vetor (): te set of all direted line segments tat are equialent to a gien direted line segment Note: Equialent means same magnitude and diretion. Component Form of a Vetor:, x x y y Ex: Grap te etor 3, and find te magnitude. One possile grap: Note: 3, ould e plaed anywere on te oordinate grid. We ae plaed it in standard position, wi is wit te initial point at te origin. Magnitude: Note: If a etor u is written in omponent form, u u1, u, ten te magnitude of u is u u u. Tis is eause te initial point is te origin, 1 0,0. Vetor ddition: Let u u1, u and 1,. Ten u1 1, u. Salar Multipliation: Let u u1, u and k e any onstant. Ten ku ku1, ku. Page 7 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

8 Note: If k 0, ten ku is in te opposite diretion. Ex: Use te grap of te etors to omplete ea example elow. w u Sow tat 1. Sow tat u. u. u Sow tat te diretion of u is te same as te diretion of. Use slope: Diretion of u = ; diretion of = Te diretion and magnitude are te same, so u. Component form of u: Component form of w:. Find te omponent form and te magnitude of uand w. u u (see aoe) w w u3w 3. Find te omponent form of u 3w. Unit Vetor: a etor wit a magnitude of 1 unit etor in te diretion of a etor an e found y diiding y te magnitude of. Unit Vetor in te Diretion of : Standard Unit Vetors: unit etors i and j in standard position along te positie x- and y-axes i 1,0 & j 0,1 ny etor an e written in terms of te standard unit etors. Ex: Write te etor,5 in terms of te standard unit etors. Ex: Find a unit etor in te diretion of te gien etor. Verify your answer is a unit etor and gie your answer in omponent form and standard unit etor form. i 4j Page 8 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

9 Find te magnitude: i4j Diide te original etor y its magnitude: i 4j 5 Component Form: Verify magnitude of unit etor: 5 5, 5 5 Reall: In te unit irle, xos, y sin. Tis leads into anoter way of expressing a etor, in terms of its diretion angle, θ. Diretion ngle: in standard position, te angle te etor makes wit te positie x-axis (ounterlokwise) Resoling a Vetor: in terms of its diretion angle, θ, a etor an e written as u os,sin u os i u sin j Ex: Find te magnitude and diretion angle of i 6j. Magnitude: Diretion angle: osi sinj ut sine we know i 6j is in Quadrant II, Page 9 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

10 Ex: Find te omponent form of gien its magnitude and its diretion angle. 5, 30 osi sinj ppliation: Resultant Fore Ex: Two fores at on an ojet: u 3, 45 and 4, 30 magnitude of te resultant fore. u u. Find te diretion and Write ea etor in omponent form: os i sin j Te resultant fore is te sum u : u ppliation: earing Ex: plane flies due east at 500 km/ and tere is a 60 km/ wit a earing of 45. Find te ground speed and te atual earing of te plane. 60 km/r 45 θ 500 km/r Sket a diagram: w p Find te etors p and w: p w Note: Te 45 is te diretion angle, not te earing. Vetor is te sum p + w: Te seond omponent of etor must equal zero, eause te plane is eaded due east. 60sin sin 0 earing of te plane: 90 Ground speed of te plane: You Try: Find te omponent form of gien its magnitude and te angle it makes wit te positie x- axis., diretion: i 3j OD: In te examples in your notes, we used sine or osine to find te diretion angle of a etor. Explain ow you ould use tangent to find te diretion angle. Page 10 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

11 Date: 6.4 Vetors and Dot Produt Syllaus Ojetie: 5.3 Te student will explore metods of etor multipliation. 5.5 Te student will determine if two etors are parallel or perpendiular (ortogonal). 5.6 Te student will derie an equation of a line or plane y using etor operations. 5.7 Te student will sole real-world appliation prolems using etors in two and tree dimensions. Dot Produt: Let u u1, u and 1,. Te dot produt is u u1 1 u. Note: Te dot produt is a salar. Ex: Ealuate 5, 3, 4. Properties of te Dot Produt: 1. u. u u = u 3. 0 u w = u w 5. u Ex: Ealuate te following gien u 3, 6 ; 1, 0 ; w 5, (a) ww () w () w u (d) u w u ngle etween Two Vetors: 1 os os Proof: Use te triangle. Law of Cosines: Property of Dot Produt: Expand: θ u u u u os - - u u os u u u u os Page 11 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

12 Property of Dot Produt: u u os Property of Equality: os os u os Ex: Find u, were θ is te angle etween u and. u 5 u 6, 8, 6 ppliation Ex: Find te interior angles of te triangle wit erties 3,0, 4,, 5,1. C os os C ngles: Ortogonal Vetors: two etors wose dot produt is equal to 0 Wat is te angle etween two non-zero ortogonal etors? u 0 os os os 0 90 Note: If te angle etween te etors is 90, we may also say tey are perpendiular. Te word ortogonal is used instead for etors eause te zero etor is ortogonal to any oter etor, ut is not perpendiular. Wat is te dot produt of two etors tat are parallel? Te angle etween tem would ae to e eiter 180 or 360. Page 1 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

13 os os180 1 u or os os360 1 u Parallel Vetors: two etors wose dot produt is equal to 1 or 1 Find w: Ex: re te etors ortogonal, parallel, or neiter? 3i j, w 3i 4j Te etors are. u Vetor Projetion: te projetion of u onto is denoted y: proj u u = u u 1 u 1 = proj u Ex: Find te projetion of onto w. Ten write as te sum of two ortogonal etors, wit one te proj w. 1,3 ; w 1,1 proj w w w w proj w ppliation: Fore Ex: Find te fore required to keep a 00-l art from rolling down a 30 inline. Draw a diagram and lael: Te fore due to graity: g00j (graity ats ertially downward) 3 1 i j i j Inline etor: os30 sin 30 f Page 13 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

14 Fore etor required to keep te art from rolling: g f projg g f Magnitude of Fore: f ppliation: Work W os foredistane Ex: person pulls a wagon wit a onstant fore of 15 ls at a onstant angle of 40 for 500 ft. Wat is te person s work? 40 w os ls 500 ft You Try: Find te projetion of onto u. Ten write as te sum of two ortogonal etors, wit one te proj. i 3 j; u i j QOD: If u is a unit etor, wat is uu? Explain wy. Page 14 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

15 Syllaus Ojeties: 7.1 Te student will grap a omplex numer on te omplex/rgand plane. 7. Te student will represent a omplex numer in trigonometri (polar) form. 7.3 Te student will simplify expressions inoling omplex numers in trigonometri (polar) form. 7.4 Te student will ompute te powers of omplex numers using DeMoire s Teorem and find te nt roots of a omplex numer. Complex Numer Plane (rgand Plane): orizontal axis real axis; ertial axis imaginary axis a+i Plotting Points in te Complex Plane Ex: Plot te points 3 4 i, 1 3 i, & i in te omplex plane. C solute Value (Modulus) of a Complex Numer: te distane a omplex numer is from te origin on te omplex plane a i a (Tis an e sown using te Pytagorean Teorem.) Ex: Ealuate 3 i. Reall: Trigonometri form of a etor: u os,sin Trigonometri Form of a Complex Numer z = a + i: z r os isin Note: Tis an also e written as z ris. a r os, rsin, r a tan a r = modulus; θ = argument Writing a Complex Numer in Trig Form Ex: Find te trigonometri form of 3i. 1. Find r: r a. Find θ: tan tan a 3. Find Quadrant 4. z r(os isin ) Page 15 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

16 Writing a Complex Numer in Standard Form (a + i) Ex: Write 9is in standard form. Expand: 9is 9os isin Multiplying and Diiding Complex Numers Let z r os isin and z r os isin Multipliation: z z r r os i sin z z r r 1 1 Diision: os i sin 1 1 Ex: Express te produt of z 1 and z in standard form. n Powers of a Complex Numer: De Moire s Teorem os sin n n z r i r os n i sin n Ex: Ealuate i 5. Rewrite in trig form: n t Roots of a Complex Numer: n n k k n k z r os i sin ris n n n n n Note: Eery omplex numer as a total of n n t roots. Ex: Find te ue roots of 8i. Write in trig form: Ealuate te roots:, k0,1,,... n 1 Page 16 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6

17 Roots of Unity: te n t roots of 1 Ex: Express te fift roots of unity in standard form and grap tem in te omplex plane. 5 t Roots of Unity: 5 1 0i r 1, 0 1is k k 1is0 1is is 5 5 You Try: 1. Write ea omplex numer in trigonometri form. Ten find te produt and te quotient. 1 3 i, 3i. Sole te equation 4 x 1 0. (You sould ae 4 solutions!) QOD: Is te trigonometri form of a omplex numer unique? Explain. Page 17 of 17 Prealulus Grapial, Numerial, lgerai: Larson Capter 6